From 7f9dbd5c263152fd3bc1218f6defa970140596b3 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 21 Dec 2020 12:44:33 +0100 Subject: Added Jorge's 157 words on why we should be in PRL. --- why_prl.txt | 21 +++++++++++++++++++++ 1 file changed, 21 insertions(+) create mode 100644 why_prl.txt diff --git a/why_prl.txt b/why_prl.txt new file mode 100644 index 0000000..48d3544 --- /dev/null +++ b/why_prl.txt @@ -0,0 +1,21 @@ + In our paper we study the extension to complex variables of the +paradigmatic +model of "complex landscape". We believe it is the first paper to study +such complex +"rugged landscapes", a subject of very high interest whose applications +range from deep networks +to optimization. In particular, we introduce and study a matrix model that +has not, to the best of our knowledge +studied previously, and which plays for the fluctuations around complex +saddles +in disordered systems the role played by the well-known semicircle law in +the real ones. +Our work is in line with the beautiful +E *Bogomolny*, O Bohigas, P *Leboeuf* - Physical Review Letters, 1992 +which concerns the roots one random polynomial of high degree, while ours +many of low degree. +We are very sure that sooner or later many new applications of this problem +will appear, as always +has been the case of extending into the complex plane the vision of a real +problem. + -- cgit v1.2.3-54-g00ecf From c93686b645159f8ed0112c38b439606503d04cd4 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 21 Dec 2020 12:45:16 +0100 Subject: Revised our 'why' statement to 100 words. --- why_prl.txt | 30 +++++++++--------------------- 1 file changed, 9 insertions(+), 21 deletions(-) diff --git a/why_prl.txt b/why_prl.txt index 48d3544..8c0eef1 100644 --- a/why_prl.txt +++ b/why_prl.txt @@ -1,21 +1,9 @@ - In our paper we study the extension to complex variables of the -paradigmatic -model of "complex landscape". We believe it is the first paper to study -such complex -"rugged landscapes", a subject of very high interest whose applications -range from deep networks -to optimization. In particular, we introduce and study a matrix model that -has not, to the best of our knowledge -studied previously, and which plays for the fluctuations around complex -saddles -in disordered systems the role played by the well-known semicircle law in -the real ones. -Our work is in line with the beautiful -E *Bogomolny*, O Bohigas, P *Leboeuf* - Physical Review Letters, 1992 -which concerns the roots one random polynomial of high degree, while ours -many of low degree. -We are very sure that sooner or later many new applications of this problem -will appear, as always -has been the case of extending into the complex plane the vision of a real -problem. - +We extend the paradigmatic model of "complex" landscapes to complex variables. +We believe it is the first such study, a subject of interest with applications +from deep networks to optimization. In particular, we introduce an apparently +new matrix model that generalizes the well-known semicircle law for +fluctuations around real saddles in disordered systems. Our work is in line +with Bogomolny, Bohigas & Leboeuf (PRL 1992) concerning roots of one random +polynomial of high degree, while ours many of low degree. Many applications of +this problem will surely appear, as always occurs when extending a real problem +into the complex plane. -- cgit v1.2.3-54-g00ecf From dd2e5767e8b7e63c5210fb4dad3ad5b5cf6fff81 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 24 Dec 2020 09:50:54 +0100 Subject: Removed incorrectly placed bibliographystyle directive and fixed capitalization of titles. --- bezout.bib | 8 ++++---- bezout.tex | 1 - 2 files changed, 4 insertions(+), 5 deletions(-) diff --git a/bezout.bib b/bezout.bib index 022f5ea..f551ece 100644 --- a/bezout.bib +++ b/bezout.bib @@ -13,7 +13,7 @@ @article{Antenucci_2015_Complex, author = {Antenucci, F. and Crisanti, A. and Leuzzi, L.}, - title = {Complex spherical {$2+4$} spin glass: A model for nonlinear optics in random media}, + title = {Complex spherical $2+4$ spin glass: A model for nonlinear optics in random media}, journal = {Physical Review A}, publisher = {American Physical Society (APS)}, year = {2015}, @@ -92,7 +92,7 @@ @article{Bray_2007_Statistics, author = {Bray, Alan J. and Dean, David S.}, - title = {Statistics of Critical Points of Gaussian Fields on Large-Dimensional Spaces}, + title = {Statistics of Critical Points of {Gaussian} Fields on Large-Dimensional Spaces}, journal = {Physical Review Letters}, publisher = {American Physical Society (APS)}, year = {2007}, @@ -148,7 +148,7 @@ @article{Crisanti_1995_Thouless-Anderson-Palmer, author = {Crisanti, A. and Sommers, H.-J.}, - title = {Thouless-Anderson-Palmer Approach to the Spherical p-Spin Spin Glass Model}, + title = {{Thouless}-{Anderson}-{Palmer} Approach to the Spherical p-Spin Spin Glass Model}, journal = {Journal de Physique I}, publisher = {EDP Sciences}, year = {1995}, @@ -176,7 +176,7 @@ @article{Dyson_1962_A, author = {Dyson, Freeman J.}, - title = {A Brownian-Motion Model for the Eigenvalues of a Random Matrix}, + title = {A {Brownian}-Motion Model for the Eigenvalues of a Random Matrix}, journal = {Journal of Mathematical Physics}, publisher = {AIP Publishing}, year = {1962}, diff --git a/bezout.tex b/bezout.tex index ff6fcdc..9e701f5 100644 --- a/bezout.tex +++ b/bezout.tex @@ -455,7 +455,6 @@ crucial role as it does in the real case. JK-D and JK are supported by the Simons Foundation Grant No.~454943. \end{acknowledgments} -\bibliographystyle{apsrev4-2} \bibliography{bezout} \end{document} -- cgit v1.2.3-54-g00ecf From d9701957da92a97eda685ac44864b12d665a285d Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Tue, 29 Dec 2020 17:20:01 +0000 Subject: Update on Overleaf. --- bezout.bib | 53 +++++++++++++++++++++++++++++++++++++++++++++++++++++ bezout.tex | 50 ++++++++++++++++++++++++++++++++------------------ 2 files changed, 85 insertions(+), 18 deletions(-) diff --git a/bezout.bib b/bezout.bib index f551ece..f791842 100644 --- a/bezout.bib +++ b/bezout.bib @@ -283,3 +283,56 @@ } +@article{cristoforetti2012new, + title={New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble}, + author={Cristoforetti, Marco and Di Renzo, Francesco and Scorzato, Luigi and AuroraScience Collaboration and others}, + journal={Physical Review D}, + volume={86}, + number={7}, + pages={074506}, + year={2012}, + publisher={APS} +} +@article{tanizaki2017gradient, + title={Gradient flows without blow-up for Lefschetz thimbles}, + author={Tanizaki, Yuya and Nishimura, Hiromichi and Verbaarschot, Jacobus JM}, + journal={Journal of High Energy Physics}, + volume={2017}, + number={10}, + pages={100}, + year={2017}, + publisher={Springer} +} + + +@article{scorzato2015lefschetz, + title={The Lefschetz thimble and the sign problem}, + author={Scorzato, Luigi}, + journal={arXiv preprint arXiv:1512.08039}, + year={2015} +} + +@article{witten2010new, + title={A new look at the path integral of quantum mechanics}, + author={Witten, Edward}, + journal={arXiv preprint arXiv:1009.6032}, + year={2010} +} + +@article{witten2011analytic, + title={Analytic continuation of Chern-Simons theory}, + author={Witten, Edward}, + journal={AMS/IP Stud. Adv. Math}, + volume={50}, + pages={347}, + year={2011} +} + +@article{behtash2015toward, + title={Toward Picard-Lefschetz theory of path integrals, complex saddles and resurgence}, + author={Behtash, Alireza and Dunne, Gerald V and Sch{\"a}fer, Thomas and Sulejmanpasic, Tin and Unsal, Mithat}, + journal={arXiv preprint arXiv:1510.03435}, + year={2015} +} + + diff --git a/bezout.tex b/bezout.tex index 9e701f5..8711dd6 100644 --- a/bezout.tex +++ b/bezout.tex @@ -72,26 +72,40 @@ constraint remains $z^2=N$. The motivations for this paper are of two types. On the practical side, there are indeed situations in which complex variables appear naturally in disordered -problems: such is the case in which they are \emph{phases}, as in random laser +problems: such is the case in which the variables are \emph{phases}, as in random laser problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used to model black hole horizons in the zero-temperature limit---also have a Hamiltonian very close to ours \cite{Anninos_2016_Disordered}. - -There is however a more fundamental reason for this study: we know from +A second reason is that we know from experience that extending a real problem to the complex plane often uncovers -underlying simplicity that is otherwise hidden. Consider, for example, the -procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying -$\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ -to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ is a -polynomial of degree $p$ chosen to have simple, known saddles. Because we are -working in complex variables, and the saddles are simple all the way (we shall -confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, -while with real variables minima of functions appear and disappear, and this -procedure is not possible. The same idea may be implemented by performing -diffusion in the $J$s and following the roots, in complete analogy with Dyson's -stochastic dynamics \cite{Dyson_1962_A}. - -The spherical constraint is enforced using the method of Lagrange multipliers: +underlying simplicity that is otherwise hidden, and thus sheds light on the original real problem +(think, for example, in the radius of convergence of a series). + +Deforming a real integration in $N$ variables to a surface of dimension $N$ in +the $2N$ dimensional complex space has turned out to be necessary for correctly defining and analyzing path integrals with complex action (see \cite{witten2010new,witten2011analytic}), and as a useful palliative for the sign-problem \cite{cristoforetti2012new,tanizaki2017gradient,scorzato2015lefschetz}. +In order to do this correctly, the features of landscape of the action in complex space must be understood. Such landscapes are in general not random: here we propose to follow the strategy of Computer Science of understanding the generic features of random instances, expecting that this sheds light on the practical, nonrandom problems. + +%Consider, for example, the +%procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying +%$\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ +%There is however a more fundamental reason for this study: +%we know from experience that extending a real problem to +%the complex plane often uncovers underlying simplicity that +%is otherwise hidden. Consider, for example, the procedure of +% +%$\lambda H_{00} + (1-\lambda) H_0$ evolving adiabatically from $\lambda=1$ to $\lambda=0$, as +%is familiar from quantum annealing. The $H_{00}$ is a polynomial +%of degree N chosen to have simple, known saddles. Because we +%are +%working in complex variables, and the saddles are simple all the way (we shall +%confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, +%while with real variables minima of functions appear and disappear, and this +%procedure is not possible. The same idea may be implemented by performing +%diffusion in the $J$s and following the roots, in complete analogy with Dyson's +%stochastic dynamics \cite{Dyson_1962_A}. + +Returning to our problem, +the spherical constraint is enforced using the method of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is \begin{equation} \label{eq:constrained.hamiltonian} H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). @@ -444,9 +458,9 @@ the complex case. The relationship between the threshold, i.e., where the gap appears, and the dynamics of, e.g., a minimization algorithm or physical dynamics, are a problem we hope to address in future work. -This paper provides a first step for the study of a complex landscape with + This paper provides a first step towards the study of a complex landscape with complex variables. The next obvious one is to study the topology of the -critical points and gradient lines of constant phase. We anticipate that the +critical points, their basins of attraction following gradient ascent (the Lefschetz thimbles), and descent (the anti-thimbles) \cite{witten2010new,witten2011analytic,cristoforetti2012new,behtash2015toward,scorzato2015lefschetz}, that act as constant-phase integrating `contours'. Locating and counting the saddles that are joined by gradient lines -- the Stokes points, that play an important role in the theory -- is also well within reach of the present-day spin-glass literature techniques. We anticipate that the threshold level, where the system develops a mid-spectrum gap, will play a crucial role as it does in the real case. -- cgit v1.2.3-54-g00ecf From 7bc5969c319a760c3259455e5bd24d1694ba1def Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 29 Dec 2020 19:15:47 +0100 Subject: Fixed citations. --- bezout.bib | 129 +++++++++++++++++++++++++++++++++++++------------------------ bezout.tex | 4 +- 2 files changed, 81 insertions(+), 52 deletions(-) diff --git a/bezout.bib b/bezout.bib index f791842..c654568 100644 --- a/bezout.bib +++ b/bezout.bib @@ -13,7 +13,7 @@ @article{Antenucci_2015_Complex, author = {Antenucci, F. and Crisanti, A. and Leuzzi, L.}, - title = {Complex spherical $2+4$ spin glass: A model for nonlinear optics in random media}, + title = {Complex spherical {$2+4$} spin glass: A model for nonlinear optics in random media}, journal = {Physical Review A}, publisher = {American Physical Society (APS)}, year = {2015}, @@ -53,6 +53,19 @@ doi = {10.1214/13-aop862} } +@article{Behtash_2017_Toward, + author = {Behtash, Alireza and Dunne, Gerald V. and Schäfer, Thomas and Sulejmanpasic, Tin and Ünsal, Mithat}, + title = {Toward {Picard}–{Lefschetz} theory of path integrals, complex saddles and resurgence}, + journal = {Annals of Mathematical Sciences and Applications}, + publisher = {International Press of Boston}, + year = {2017}, + number = {1}, + volume = {2}, + pages = {95--212}, + url = {https://doi.org/10.4310%2Famsa.2017.v2.n1.a3}, + doi = {10.4310/amsa.2017.v2.n1.a3} +} + @book{Bezout_1779_Theorie, author = {Bézout, Etienne}, title = {Théorie générale des équations algébriques}, @@ -160,6 +173,20 @@ doi = {10.1051/jp1:1995164} } +@article{Cristoforetti_2012_New, + author = {Cristoforetti, Marco and Di Renzo, Francesco and Scorzato, Luigi}, + title = {New approach to the sign problem in quantum field theories: High density {QCD} on a {Lefschetz} thimble}, + journal = {Physical Review D}, + publisher = {American Physical Society (APS)}, + year = {2012}, + month = {10}, + number = {7}, + volume = {86}, + pages = {074506}, + url = {https://doi.org/10.1103%2Fphysrevd.86.074506}, + doi = {10.1103/physrevd.86.074506} +} + @article{Cugliandolo_1993_Analytical, author = {Cugliandolo, L. F. and Kurchan, J.}, title = {Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model}, @@ -268,6 +295,33 @@ doi = {10.2307/2371510} } +@inproceedings{Scorzato_2016_The, + author = {Scorzato, Luigi}, + title = {The {Lefschetz} thimble and the sign problem}, + publisher = {Sissa Medialab}, + year = {2016}, + month = {7}, + volume = {251}, + url = {https://doi.org/10.22323%2F1.251.0016}, + doi = {10.22323/1.251.0016}, + booktitle = {Proceedings of The 33rd International Symposium on Lattice Field Theory (LATTICE 2015)}, + series = {Proceedings of Science} +} + +@article{Tanizaki_2017_Gradient, + author = {Tanizaki, Yuya and Nishimura, Hiromichi and Verbaarschot, Jacobus J. M.}, + title = {Gradient flows without blow-up for {Lefschetz} thimbles}, + journal = {Journal of High Energy Physics}, + publisher = {Springer Science and Business Media LLC}, + year = {2017}, + month = {10}, + number = {10}, + volume = {2017}, + pages = {100}, + url = {https://doi.org/10.1007%2Fjhep10%282017%29100}, + doi = {10.1007/jhep10(2017)100} +} + @article{Weyl_1912_Das, author = {Weyl, Hermann}, title = {Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)}, @@ -282,57 +336,32 @@ doi = {10.1007/bf01456804} } - -@article{cristoforetti2012new, - title={New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble}, - author={Cristoforetti, Marco and Di Renzo, Francesco and Scorzato, Luigi and AuroraScience Collaboration and others}, - journal={Physical Review D}, - volume={86}, - number={7}, - pages={074506}, - year={2012}, - publisher={APS} -} -@article{tanizaki2017gradient, - title={Gradient flows without blow-up for Lefschetz thimbles}, - author={Tanizaki, Yuya and Nishimura, Hiromichi and Verbaarschot, Jacobus JM}, - journal={Journal of High Energy Physics}, - volume={2017}, - number={10}, - pages={100}, - year={2017}, - publisher={Springer} -} - - -@article{scorzato2015lefschetz, - title={The Lefschetz thimble and the sign problem}, - author={Scorzato, Luigi}, - journal={arXiv preprint arXiv:1512.08039}, - year={2015} -} - -@article{witten2010new, - title={A new look at the path integral of quantum mechanics}, - author={Witten, Edward}, - journal={arXiv preprint arXiv:1009.6032}, - year={2010} -} - -@article{witten2011analytic, - title={Analytic continuation of Chern-Simons theory}, - author={Witten, Edward}, - journal={AMS/IP Stud. Adv. Math}, - volume={50}, - pages={347}, - year={2011} +@article{Witten_2010_A, + author = {Witten, Edward}, + title = {A new look at the path integral of quantum mechanics}, + journal = {Surveys in Differential Geometry}, + publisher = {International Press of Boston}, + year = {2010}, + number = {1}, + volume = {15}, + pages = {345--420}, + url = {https://doi.org/10.4310%2Fsdg.2010.v15.n1.a11}, + doi = {10.4310/sdg.2010.v15.n1.a11} } -@article{behtash2015toward, - title={Toward Picard-Lefschetz theory of path integrals, complex saddles and resurgence}, - author={Behtash, Alireza and Dunne, Gerald V and Sch{\"a}fer, Thomas and Sulejmanpasic, Tin and Unsal, Mithat}, - journal={arXiv preprint arXiv:1510.03435}, - year={2015} +@incollection{Witten_2011_Analytic, + author = {Witten, Edward}, + title = {Analytic continuation of {Chern}-{Simons} theory}, + publisher = {American Mathematical Society}, + year = {2011}, + month = {7}, + volume = {50}, + pages = {347--446}, + url = {https://doi.org/10.1090%2Famsip%2F050%2F19}, + doi = {10.1090/amsip/050/19}, + booktitle = {Chern-Simons Gauge Theory: 20 Years After}, + editor = {Andersen, Jørgen E. and Boden, Hans U. and Hahn, Atle and Himpel, Benjamin}, + series = {AMS/IP Studies in Advanced Mathematics} } diff --git a/bezout.tex b/bezout.tex index 8711dd6..d496e52 100644 --- a/bezout.tex +++ b/bezout.tex @@ -82,7 +82,7 @@ underlying simplicity that is otherwise hidden, and thus sheds light on the orig (think, for example, in the radius of convergence of a series). Deforming a real integration in $N$ variables to a surface of dimension $N$ in -the $2N$ dimensional complex space has turned out to be necessary for correctly defining and analyzing path integrals with complex action (see \cite{witten2010new,witten2011analytic}), and as a useful palliative for the sign-problem \cite{cristoforetti2012new,tanizaki2017gradient,scorzato2015lefschetz}. +the $2N$ dimensional complex space has turned out to be necessary for correctly defining and analyzing path integrals with complex action (see \cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the sign-problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient, Scorzato_2016_The}. In order to do this correctly, the features of landscape of the action in complex space must be understood. Such landscapes are in general not random: here we propose to follow the strategy of Computer Science of understanding the generic features of random instances, expecting that this sheds light on the practical, nonrandom problems. %Consider, for example, the @@ -460,7 +460,7 @@ dynamics, are a problem we hope to address in future work. This paper provides a first step towards the study of a complex landscape with complex variables. The next obvious one is to study the topology of the -critical points, their basins of attraction following gradient ascent (the Lefschetz thimbles), and descent (the anti-thimbles) \cite{witten2010new,witten2011analytic,cristoforetti2012new,behtash2015toward,scorzato2015lefschetz}, that act as constant-phase integrating `contours'. Locating and counting the saddles that are joined by gradient lines -- the Stokes points, that play an important role in the theory -- is also well within reach of the present-day spin-glass literature techniques. We anticipate that the +critical points, their basins of attraction following gradient ascent (the Lefschetz thimbles), and descent (the anti-thimbles) \cite{Witten_2010_A, Witten_2011_Analytic, Cristoforetti_2012_New, Behtash_2017_Toward, Scorzato_2016_The}, that act as constant-phase integrating `contours'. Locating and counting the saddles that are joined by gradient lines -- the Stokes points, that play an important role in the theory -- is also well within reach of the present-day spin-glass literature techniques. We anticipate that the threshold level, where the system develops a mid-spectrum gap, will play a crucial role as it does in the real case. -- cgit v1.2.3-54-g00ecf From 9cec3f1cff603521aa4e948576b56f06d5902f39 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 29 Dec 2020 19:46:57 +0100 Subject: English and paragraph quadriture. --- bezout.tex | 89 +++++++++++++++++++++++++++++--------------------------------- 1 file changed, 41 insertions(+), 48 deletions(-) diff --git a/bezout.tex b/bezout.tex index d496e52..9dffe65 100644 --- a/bezout.tex +++ b/bezout.tex @@ -72,49 +72,37 @@ constraint remains $z^2=N$. The motivations for this paper are of two types. On the practical side, there are indeed situations in which complex variables appear naturally in disordered -problems: such is the case in which the variables are \emph{phases}, as in random laser -problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used to model -black hole horizons in the zero-temperature limit---also have a Hamiltonian -very close to ours \cite{Anninos_2016_Disordered}. -A second reason is that we know from -experience that extending a real problem to the complex plane often uncovers -underlying simplicity that is otherwise hidden, and thus sheds light on the original real problem -(think, for example, in the radius of convergence of a series). - -Deforming a real integration in $N$ variables to a surface of dimension $N$ in -the $2N$ dimensional complex space has turned out to be necessary for correctly defining and analyzing path integrals with complex action (see \cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the sign-problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient, Scorzato_2016_The}. -In order to do this correctly, the features of landscape of the action in complex space must be understood. Such landscapes are in general not random: here we propose to follow the strategy of Computer Science of understanding the generic features of random instances, expecting that this sheds light on the practical, nonrandom problems. - -%Consider, for example, the -%procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying -%$\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ -%There is however a more fundamental reason for this study: -%we know from experience that extending a real problem to -%the complex plane often uncovers underlying simplicity that -%is otherwise hidden. Consider, for example, the procedure of -% -%$\lambda H_{00} + (1-\lambda) H_0$ evolving adiabatically from $\lambda=1$ to $\lambda=0$, as -%is familiar from quantum annealing. The $H_{00}$ is a polynomial -%of degree N chosen to have simple, known saddles. Because we -%are -%working in complex variables, and the saddles are simple all the way (we shall -%confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, -%while with real variables minima of functions appear and disappear, and this -%procedure is not possible. The same idea may be implemented by performing -%diffusion in the $J$s and following the roots, in complete analogy with Dyson's -%stochastic dynamics \cite{Dyson_1962_A}. - -Returning to our problem, -the spherical constraint is enforced using the method of Lagrange multipliers: -introducing $\epsilon\in\mathbb C$, our energy is +problems: such is the case in which the variables are \emph{phases}, as in +random laser problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used +to model black hole horizons in the zero-temperature limit---also have a +Hamiltonian very close to ours \cite{Anninos_2016_Disordered}. A second reason +is that, as we know from experience, extending a real problem to the complex +plane often uncovers underlying simplicity that is otherwise hidden, sheding +light on the original real problem, e.g., as in the radius of convergence of a +series. + +Deforming an integral in $N$ real variables to a surface of dimension $N$ in +$2N$-dimensional complex space has turned out to be necessary for correctly +defining and analyzing path integrals with complex action (see +\cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the +sign problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient, +Scorzato_2016_The}. In order to do this correctly, the features of landscape +of the action in complex space---like the relative position of its +saddles---must be understood. Such landscapes are in general not random: here +we propose to follow the strategy of computer science of understanding the +generic features of random instances, expecting that this sheds light on the +practical, nonrandom problems. + +Returning to our problem, the spherical constraint is enforced using the method +of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is \begin{equation} \label{eq:constrained.hamiltonian} H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). \end{equation} - We choose to -constrain our model by $z^2=N$ rather than $|z|^2=N$ in order to preserve the -analyticity of $H$. The nonholomorphic constraint also has a disturbing lack of -critical points nearly everywhere: if $H$ were so constrained, then -$0=\partial^* H=-p\epsilon z$ would only be satisfied for $\epsilon=0$. +We choose to constrain our model by $z^2=N$ rather than $|z|^2=N$ in order to +preserve the analyticity of $H$. The nonholomorphic constraint also has a +disturbing lack of critical points nearly everywhere: if $H$ were so +constrained, then $0=\partial^* H=-p\epsilon z$ would only be satisfied for +$\epsilon=0$. The critical points are of $H$ given by the solutions to the set of equations \begin{equation} \label{eq:polynomial} @@ -122,12 +110,11 @@ The critical points are of $H$ given by the solutions to the set of equations = p\epsilon z_i \end{equation} for all $i=\{1,\ldots,N\}$, which for fixed $\epsilon$ is a set of $N$ -equations of degree $p-1$, to which one must add the constraint. -In this sense +equations of degree $p-1$, to which one must add the constraint. In this sense this study also provides a complement to the work on the distribution of zeroes of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ -and $p\to\infty$. -We see from \eqref{eq:polynomial} that at any critical point, $\epsilon=H/N$, the average energy. +and $p\to\infty$. We see from \eqref{eq:polynomial} that at any critical +point, $\epsilon=H/N$, the average energy. Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a critical point of $\operatorname{Im}H$. The number of critical points of $H$ is @@ -459,10 +446,16 @@ appears, and the dynamics of, e.g., a minimization algorithm or physical dynamics, are a problem we hope to address in future work. This paper provides a first step towards the study of a complex landscape with -complex variables. The next obvious one is to study the topology of the -critical points, their basins of attraction following gradient ascent (the Lefschetz thimbles), and descent (the anti-thimbles) \cite{Witten_2010_A, Witten_2011_Analytic, Cristoforetti_2012_New, Behtash_2017_Toward, Scorzato_2016_The}, that act as constant-phase integrating `contours'. Locating and counting the saddles that are joined by gradient lines -- the Stokes points, that play an important role in the theory -- is also well within reach of the present-day spin-glass literature techniques. We anticipate that the -threshold level, where the system develops a mid-spectrum gap, will play a -crucial role as it does in the real case. + complex variables. The next obvious one is to study the topology of the + critical points, their basins of attraction following gradient ascent (the + Lefschetz thimbles), and descent (the anti-thimbles) \cite{Witten_2010_A, + Witten_2011_Analytic, Cristoforetti_2012_New, Behtash_2017_Toward, + Scorzato_2016_The}, which act as constant-phase integrating `contours.' + Locating and counting the saddles that are joined by gradient lines---the + Stokes points, which play an important role in the theory---is also well within + reach of the present-day spin-glass literature techniques. We anticipate + that the threshold level, where the system develops a mid-spectrum gap, will + play a crucial role as it does in the real case. \begin{acknowledgments} We wish to thank Alexander Altland, Satya Majumdar and Gregory Schehr for a useful suggestions. -- cgit v1.2.3-54-g00ecf From 3aa483b31c12d6a3afb1215ce6188aca8c314e72 Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Wed, 30 Dec 2020 11:03:54 +0000 Subject: Update on Overleaf. --- bezout.bib | 7 +- bezout.tex | 6 +- cover.tex | 1358 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 1367 insertions(+), 4 deletions(-) create mode 100644 cover.tex diff --git a/bezout.bib b/bezout.bib index c654568..8688ed2 100644 --- a/bezout.bib +++ b/bezout.bib @@ -294,7 +294,12 @@ url = {https://doi.org/10.2307%2F2371510}, doi = {10.2307/2371510} } - +@book{mezard2009information, + title={Information, physics, and computation}, + author={Mezard, Marc and Montanari, Andrea}, + year={2009}, + publisher={Oxford University Press} +} @inproceedings{Scorzato_2016_The, author = {Scorzato, Luigi}, title = {The {Lefschetz} thimble and the sign problem}, diff --git a/bezout.tex b/bezout.tex index 9dffe65..96ad257 100644 --- a/bezout.tex +++ b/bezout.tex @@ -44,7 +44,7 @@ Spin-glasses have long been considered the paradigm of many variable `complex landscapes,' a subject that includes neural networks and optimization problems, -most notably constraint satisfaction. The most tractable family of these +most notably constraint satisfaction \cite{mezard2009information}. The most tractable family of these are the mean-field spherical $p$-spin models \cite{Crisanti_1992_The} (for a review see \cite{Castellani_2005_Spin-glass}) defined by the energy \begin{equation} \label{eq:bare.hamiltonian} @@ -447,8 +447,8 @@ dynamics, are a problem we hope to address in future work. This paper provides a first step towards the study of a complex landscape with complex variables. The next obvious one is to study the topology of the - critical points, their basins of attraction following gradient ascent (the - Lefschetz thimbles), and descent (the anti-thimbles) \cite{Witten_2010_A, + critical points, the sets reached following gradient descent (the + Lefschetz thimbles), and ascent (the anti-thimbles) \cite{Witten_2010_A, Witten_2011_Analytic, Cristoforetti_2012_New, Behtash_2017_Toward, Scorzato_2016_The}, which act as constant-phase integrating `contours.' Locating and counting the saddles that are joined by gradient lines---the diff --git a/cover.tex b/cover.tex new file mode 100644 index 0000000..586fa38 --- /dev/null +++ b/cover.tex @@ -0,0 +1,1358 @@ + +\documentclass[12pt,reqno,a4paper,twoside]{article} +% \ProvidesPackage{makra} +\usepackage{amsmath,amsthm,amstext,amscd,amssymb,euscript} +%,showkeys} +%,times} +\usepackage{epsf} +\usepackage{color} +\usepackage{verbatim} +\usepackage{graphicx} +\usepackage{esint} +\usepackage{tikz} +\usepackage{setspace} +\usepackage{mathrsfs} + +\usepackage{todonotes} + +%\usepackage{natbib} + + + +\usepackage{bm} +\usepackage[normalem]{ulem} + + +\textwidth 6in +\topmargin -0.50in +\oddsidemargin 0in +\evensidemargin 0in +\textheight 9.00in +%\pagestyle{plain} +%%%%%%%%%%%%%%%%%% Macros %%%%%%%%%%% +\def\mybox #1{\fbox{\parbox{5.8in}{#1}}} +\newcommand{\m}[1]{{\marginpar{\scriptsize #1}}} + +\def\mep{\mathbf{mep}_{n}^{\delta}} +\def\r{{\mathbf r}} +\def\O{{\mathcal{O}}} + +\def\I{{\mathcal{I}}} +\def\fee{\mathcal{F}} + +\def\F{{\EuScript{F}}} + +\renewcommand{\phi}{\varphi} +\newcommand{\compose}{\circ} +\renewcommand{\subset}{\subseteq} +\renewcommand{\emptyset}{\varnothing} +\newcommand{\interval}{[\underline\alpha,\overline \alpha]} +\def\liminfn{\liminf_{n\to\infty}} +\def\limsupn{\limsup_{n\to\infty}} +\def\limn{\lim_{n\to\infty}} +\def\disagree{\not\longleftrightarrow} +\newcommand{\Zd}{\mathbb Z^d} +\newcommand{\kk}{\mathbf k} +\renewcommand{\Pr}{\mathbb P} +\newcommand{\dist}{\text{dist}} +\newcommand{\Cal}{\mathcal} +\def\1{ {\mathit{1} \!\!\>\!\! 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+\filldraw[white] (4.3,4.6) circle (1pt) ; +\end{tikzpicture} +} + +\setstretch{1.24} + +\begin{document} + +\title{{\bf COVER LETTER \\`Complex complex landscapes'}} +%\footnote{{\bf Key-words}: } +%}} + +\author{ +Jaron Kent-Dobias + and +Jorge Kurchan +} + +\maketitle + + + +\vspace{1.cm} + + +The subject of `Complex Landscapes', which started in the spin-glass literature, is concerned with functions (landscapes) of many variables, having a multiplicity of minimums, which are the objects of interest. Apart from its obvious interest for glassy systems, it has found a myriad applications in many domains: Computer Science, Ecology, Economics, Biology \cite{mezard2009information}. + +In the last few years, a renewed interest has developed for landscapes for which the variables are complex. There are a few reasons for this: {\em i)} in Computational Physics, there is the main obstacle of the `sign problem', and a strategy has emerged to attack it deforming the sampling space into complex variables. This is a most natural and promising path, and any progress made will have game-changing impact in solid state physics and lattice-QCD \cite{Cristoforetti_2012_New,Scorzato_2016_The}. +{\em ii)} At a more basic level, following the seminal work of E. Witten \cite{Witten_2010_A,Witten_2011_Analytic}, there has been a flurry of activity concerning the very definition of quantum mechanics, which requires also that one move into the complex plane. + +In all these cases, just like in the real case, one needs to know the structure of the `landscape', where are the saddle points and how they are connected, typical questions of `complexity'. +However, to the best of our knowledge, there are no studies extending the methods of the theory of +complexity to +complex variables. +We believe our paper will open a field that may find +numerous applications and will widen our theoretical view of complexity in general. + + +\bibliography{bezout} + + +\end{document} + + + + + + + + + + + + + + + +\section{The Kipnis-Marchioro-Presutti model} + +Consider the following process: +\begin{itemize} +\item +choose a pair of neighbouring sites and completely +exchange energy between them +\item +if the site is one of the borders, exchange completely energy with the bath. +\end{itemize} +each choice with probability $1/(N+1)$. From here onwards, we shall denote +$\tau$ a large time, sufficient for any two-site thermalisation. + +The evolution operator in one step is: +\begin{eqnarray} +U &=& \frac{1}{N+1} \left[ e^{-\tau L_1^*} + e^{-\tau L_N^*} + \sum_{i=1}^{N-1} e^{-\tau L^*_{i,i+1}} \right] +\nonumber \\ +&=& \frac{1}{N+1} \left[ e^{-2\tau (T_1 K^-_1 + K^o_1 + k) } + e^{-2\tau(T_L K^-_L + K^o_L +k) } + + \sum_{i=1}^{N-1} e^{ \frac{-\tau}{k} +(K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1} ++ 2k^2 )} \right] \nonumber \\ +~ +\end{eqnarray} +and the dynamics after $n$ steps is given by $U^n$. +Because we are considering large $\tau$, the terms in the sums are in fact projectors +onto the lowest eigenvalues of the exponents. We shall however keep the notation as it is +in order to stress the symmetry of the bulk terms. + +Let us now show that - at the level of energies - this dynamics yields the KMP process +{\em for $k=\frac{1}{2}$, that is $m=2$}. +Consider first a general $m$, and two neighbouring sites of coordinates $x = \{x_\alpha\}_{\alpha=1,\ldots,m}$, +$y=\{y_\alpha\}_{\alpha=1,\ldots,m}$. +If they are completely thermalised, it means that (cfr (\ref{bb}): +the joint probability density satisfies +\begin{equation} +\left(x_{\alpha} +\frac{\partial}{\partial y_{\beta}} - +y_{\beta}\frac{\partial}{\partial x_{\alpha}} + \right) p(x,y)=0 +\end{equation} +It is easy to see that this may happen if and only if +\begin{equation} +p(x,y)= p[ \sum_\alpha (x_\alpha^2+y_\alpha^2)] +\end{equation} +In particular let us consider the microcanonical measure +\begin{equation} +p(x,y)= \delta[ \sum_\alpha (x_\alpha^2+y_\alpha^2)-\epsilon ] +\end{equation} +Defining new random variables $\epsilon_1$ and $\epsilon_2$ +as the energies of the neighboring sites +\be +\epsilon_1 = \sum_\alpha x_\alpha^2 +\ee +\be +\epsilon_2 = \sum_\alpha y_\alpha^2 +\ee +then their joint probability density will be +\begin{equation} +p(\epsilon_1,\epsilon_2) = \frac{S_m^2}{4} \delta(\epsilon_1+\epsilon_2-\epsilon) +\epsilon_1^{\frac{1}{2}-1} \epsilon_2^{\frac{1}{2}-1} +\end{equation} +where $S_m$ denotes the surface of the unit sphere in $m$ dimension +\be +S_m = \frac{m \pi^{m/2}}{\Gamma(\frac{1}{2}+1)} +\ee +{\em This yields a flat distribution for $m=2$, i.e. the KMP model.} + + + + +\section{Dual model} + + +The expectation value of an observable at time $t$, starting from an initial +distribution $|init\rangle$ is: + + +\begin{equation} + = \langle - | O e^{-Ht} | init \rangle +\end{equation} +where $\langle - |$ is a constant. +Taking the adjoint $ x_i \to x_i$, $\partial_i \to -\partial_i$: +\begin{equation} + = \langle - | O e^{-Ht} | init \rangle= \langle init| e^{-H^\dag t} O |- \rangle +\end{equation} +where $H^\dag(K^\pm, K^o)=H( K^\pm, -K^o)$ (because of the change of signs of the derivatives) +\begin{eqnarray} +-H^\dag&=& \frac{4}{1} \sum_i \left( +K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1} ++ \frac{m^2}{8} \right) +\nonumber\\ +&+&2 \left(T_1 K^-_1 + K^o_1 + \frac{1}{4}\right) ++2 \left(T_L K^-_L + K^o_L +\frac{1}{4}\right) +\end{eqnarray} +In particular, for the generating function we had chosen + \begin{equation} + O |- \rangle = \Pi_i \frac{x_i^{2 \xi_i}}{(2\xi_i -1)!!}|-\rangle=|\xi_1,...,\xi_N\rangle +\end{equation} + +Considered as an operator acting on `particle number', as counted by $K^o$, $H^\dag$ does not +conserve the probability. +The trick we used can be expressed as follows: introduce the particle number $\xi_o$ and $\xi_{N+1}$ +and the operators $A^+_o$ and $A^+_{N+1}$, which create particles in boundary sites with unit rate. +We consider now the {\em enlarged} process generated by +\begin{eqnarray} +-H^{dual}&=& \frac{4}{1} \sum_i \left( +K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1} ++ \frac{m^2}{8} \right) +\nonumber\\ +&+&2 \left(A^+_o K^-_1 + K^o_1 - \frac{1}{4}\right) ++2 \left( A^+_{N+1} K^-_N + K^o_N -\frac{1}{4}\right) +\end{eqnarray} +which conserves ({\it seems}) particle number and probability. +We wish to prove that: + +\begin{eqnarray} + &=& \langle init| e^{-H^\dag t} |\xi_1,...,\xi_N \rangle \nonumber \\ +&=& \sum_{\xi_o,\xi_{N+1}} + T_1^{\xi_o} T_{L}^{\xi_{N+1}} \langle \xi_o \xi_{N+1} | \otimes \langle + init| e^{-H^{dual} t} |\xi_1,...,\xi_N \rangle \otimes + |\xi_o=0,\xi_{N+1}=0 \rangle \nonumber \\ +\label{ggg} +\end{eqnarray} + + +I think the proof is obvious, because developing the exponential of $H^{dual}$ all the $A^+$ can be +collected because they commute with everything else, and the experctation value +\begin{equation} +\sum_{\xi_o} T_1^{\xi_o} \langle \xi_o |[A^+_o]^r |\xi_o=0 \rangle = T_1^r + \end{equation} +just puts back as many $T$'s as necessary. + +I do not know exactly how to use (\ref{ggg}) in general, but in the large time limit the evolution +voids the chain of particles + + +\section{Dual of KMP} + +I think that the argument runs through without changes if we use $U$ defined for the KMP model. +We just have to note that each term corresponds to an evolution of two sites (or a site and the bath) +and so in the dual it corresponds to sharing the particles between those two sites, or emptying +the sites at the borders. + +{\bf: NOTE by Cristian} + +We can check that the duality function chosen in the original paper by KMP +do coincide with the duality function of our process for $m=2$ (and the random +variables are the energies). +Indeed we start from +\be +f(x,\xi) = \prod_i (\sum_{\alpha} x_{i,\alpha}^2)^{\xi} +\ee +When the bath have equal temperature (let's us choose T=1) then the stationary +measure is +\be +\pi(x) = \prod_i \frac{1}{(2\pi)^{m/2}} \exp\left(-\sum_{\alpha}\frac{x_{i,\alpha}^2}{2}\right) +\ee +Let us focus on a fixed $i$ (that is in this short computation we write $x$ for $x_i$). +We have +\begin{eqnarray} +\E(f(x,\xi)) +&=& +\int dx_1 \cdots \int dx_m (x_1^2+\ldots + x_m^2)^{\xi} \exp-\left(\frac{x_{1}^2}{2}+\ldots+\frac{x_{1}^2}{2}\right) +\nonumber \\ +& = & +\int dr S_m r^{2\xi} \exp-\left(\frac{r^2}{2}\right) +\nonumber \\ +& = & +\frac{\frac{1}{2}\Gamma(\frac{1}{2}+\xi)}{\Gamma(\frac{m}{2}+1)} 2^\xi +\nonumber \\ +\end{eqnarray} +Special cases: +\begin{itemize} +\item $m=1$ + +$$ +\E(f(x,\xi)) = (2\xi-1)!! +$$ +where one uses that $\Gamma(\frac{1}{2}+\xi)= \frac{\sqrt{\pi}(2\xi-1)!!}{2^{\xi}}$ and $\Gamma(\frac{3}{2}) = \frac{\sqrt{\pi}}{2}$ +\item $m=2$ + +$$ +\E(f(x,\xi)) = \xi! 2^\xi +$$ +where one uses that $\Gamma(1+\xi)= \xi!$ and $\Gamma(2) = 1$. +Thus, if one defines the energies as +$$ +\epsilon_i = \sum_{\alpha}\frac{x_{i,\alpha}^2}{2} +$$ +one recover the choice of KMP for the dual function +$$ +O(\epsilon_i,\xi) = \prod_i \frac{\epsilon_i^{\xi_i}}{\xi_i!} +$$ +\end{itemize} + + + + + + +\section{ Dual of SEP: here goes an outline of how to proceed for the SSEP} + + +\be +H=-L_{SEP}^* +\ee +\begin{eqnarray} +L^*_{SEP} &=& \frac{1}{j} + \sum_i \left(J^+_i J^-_{i+1} + J^-_i J^+_{i+1} + 2 J^o_i J^o_{i+1} + - 2 j^2 \right)\\ +&+&\alpha (J^-_1 - J^o_1-j) + \gamma (J^+_1 + J^o_1-j) ++ \delta (J^-_L - J^o_L-j) + \beta (J^+_L + J^o_L-j)\nonumber +\end{eqnarray} +The factor $1/j$ is analogous to the factor $1/m$ in (\ref{bb}). +The operators $J^+_i, J^-_i, J^o_i$ act on the Hilbert space + corresponding to $0 \le r \le n$ particles per site $\otimes_i |r\rangle_i$ +as follows: +\begin{eqnarray} +J^+_i |r\rangle_i &=& (2j-r) |r+1\rangle_i \nonumber \\ + J^-_i |r\rangle_i &=& r |r-1\rangle_i \nonumber \\ +J^o_i |r\rangle_i &=& (r-j) |r\rangle_i +\end{eqnarray} + +The conjugation properies are as follows. There is an operator $Q$, +{\em diagonal in this basis } (I give the expression below), such that: +\begin{equation} +[J^+_i]^\dag = Q[J^-_i]Q^{-1} \qquad [J^-_i]^\dag = Q[J^+_i]Q^{-1} +\end{equation} +while $[J^z_i]^\dag=J^z_i= Q[J^z_i]Q^{-1}$. + + + +The expectation value of an observable at time $t$, starting from an initial +distribution $|init\rangle$ is: + + +\begin{equation} + = \langle - | O e^{-Ht} | init \rangle +\end{equation} +where $\langle - |$ is a constant. +As before: +\begin{eqnarray} + &=& \langle - | O e^{-Ht} | init \rangle= +\langle init| e^{-H^\dag t} O |- \rangle= \nonumber \\ +& & \langle init|Q e^{-{\bar H} t} Q^{-1}O |- \rangle= +\langle init|Q \; e^{-{\bar H} t} Q^{-1}O Q Q^{-1} |- \rangle +\end{eqnarray} + + +{\em $ {\bar H}$ is the same operator as $H$ but with +$J^+$ substituted by $J^-$, and vice-versa.} +Our job is now to make the rotation that will eliminate the $J^+$'s in +the border terms of $ {\bar H}$. + + + + +The transformation is of the form +\begin{eqnarray} +e^{\mu J^+} J^+ e^{-\mu J^+}&=&J^+ \nonumber \\ +e^{\mu J^+} J^o e^{-\mu J^+} &=&J^o - \mu J^+ \nonumber \\ +e^{\mu J^+} J^- e^{-\mu J^+} &=& J^- + 2 \mu J^o - \mu^2 J^+ +\end{eqnarray} +for suitable $\mu$. +Putting $\mu=-1$, we get that {\bf the bulk term is left invariant, +precisely because of the SU(2) symmetry}. The boundary terms {\bf of $\bar H$} +transform further into: +\begin{eqnarray} +& e^{\mu J^+_1} \left[ \alpha (J^+_1 - J^o_1-j) + \gamma (J^-_1 + J^o_1-j) +\right] e^{-\mu J^+_1}= \nonumber \\ & \gamma(J^-_1 + 2 \mu J^o_1 - \mu^2 +J^+_1 +J^o_1 - \mu J^+_1 -j) + \alpha (J^+_1 - J^o_1 + \mu J^+_1 -j) += \nonumber \\ +& \alpha(- J^o_1 -j) + \gamma (J^-_1 -J^o_1 -j) +\label{trans} +\end{eqnarray} +which is of the same form we have in the $SU(1,1)$ model. +The same can be done in the other boundary term. + +We thus get: +\begin{eqnarray} + &=& \langle - | O e^{-Ht} | init \rangle= +\langle init|Q \; e^{-{\bar H} t} Q^{-1}O Q Q^{-1} |- \rangle \nonumber\\ +&= & \langle init|Q e^{ \sum_i J^+_i} e^{-{\bar H_{dual}} t} + e^{ -\sum_i J^+_i} Q^{-1}O Q Q^{-1} |- \rangle \nonumber \\ +&= & \langle init|Q e^{ \sum_i J^+_i} e^{-{\bar H_{dual}} t} + e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i J^+_i} + e^{ -\sum_i J^+_i} |- \rangle \nonumber \\ + &= & \langle init|Q Q^{-1} e^{ \sum_i J^+_i} e^{-{\bar H_{dual}} t} + e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i J^+_i} |-_{dual} \rangle +\end{eqnarray} +where we have defined $H_{dual}$ as the transformed Hamiltonian. + +We now have to study $ |-_{dual} \rangle \equiv e^{ -\sum_i J^+_i} + Q^{-1} |- \rangle$ +Because we know that terms like those proportional to $\gamma$ and $\alpha$ +anihilate the measure to the left: +\begin{eqnarray} +& & \langle - | (J^-_i - J^o_i-j) =0\nonumber \\ +& & \langle - | (J^+_i + J^o_i-j) =0 +\end{eqnarray} +this implies that in the new variables and following all the transformations +(cfr (\ref{trans})): +\begin{eqnarray} +& & (J^-_i -J^o_i -j)e^{ -\sum_i J^+_i} Q^{-1} |- \rangle= 0 \nonumber \\ +& & ( -J^o_i -j)e^{ -\sum_i J^+_i} Q^{-1} |- \rangle =0 +\end{eqnarray} +which implies that $( J^o_i +j) |-_{dual} \rangle= J^-_i |-_{dual} \rangle=0$, +and this means that +\begin{equation} +J^o_i |-_{dual} \rangle =-j |-_{dual} \rangle +\end{equation} +is the vacuum of particles in this base! + +All in all we are left with: +\begin{eqnarray} + &=& \langle init|Q \; e^{ \sum_i J^+_i} + e^{-{\bar H_{dual}} t} e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i J^+_i} + |-_{dual} \rangle \nonumber \\ + &=& \langle init|Q \; e^{ \sum_i J^+_i} + e^{-{\bar H_{dual}} t} {\hat O} + |-_{dual} \rangle +\end{eqnarray} +where $ {\hat O} \equiv e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i +J^+_i}$. We have to start with the +vacuum $ |-_{dual} \rangle$, then apply $ {\hat O} $, (which creates +particles because it contains many $J^+$'s), and then there is the +dual evolution. The final configuration has to be overlapped with +$\langle f| \equiv \langle init|Q \; e^{ \sum_i J^+_i}$. +For large times, there will be no particle left except in the two extra sites +in the borders. + +\section{Constructive approach} + +Here I would like to say the following: if I have a modle of transport +of which I do not know if it has a Dual one, I can proceed as follows. +I take a small version with no baths and a few sites. I write the +evolution operator and I diagonalise it numerically. If there is a +non abelian group, the eigenvalues will be in degenerate +multiplets. Hence, if I find multiplets, then very probably there is a +dual model, if I do not, then there cannot be one. It would be nice +to show it with the KMP model with two or three sites. + +Another thing is to consider higher groups. $SU(3)$ has already been studied +for two kinds of particles. We know how to map to a dual in that +case, if it has not been done yet. + +\newpage +{\bf THIS PART HAS BEEN WRITTEN BY CRISTIAN} + +The aim of this file is to set notation in the two languages. +Let us focus on duality for the case we already know: +SU(1,1) model with $k=1/4$. To fix ideas let us consider only +the bulk part of the system with periodic boundary conditions. + +\section{Probabilistic language} +We have two stochastic Markovian process with continuous time. +\begin{itemize} +\item +\underline{The first process $X(t) \in \R^N$} is given by the Fokker-Planck equation: +\be +\frac{dp(x,t)}{dt} = L^* p(x,t) +\ee +where $p(x,t)$ represents the probability density +for the process $X(t)$, that is +$$ +p(x,t)dx = Prob (X(t)\in (x,x+dx)) +$$ +and +\begin{eqnarray} +L^*p(x,t) +& = & +\sum_i L^*_{i,i+1} p(x,t) \noindent\\ +& = & +\sum_i \left(x_i\frac{\partial}{\partial x_{i+1}} -x_{i+1}\frac{\partial}{\partial x_{i}}\right)^2 p(x,t) +\end{eqnarray} +\item +\underline{The second process $\Xi(t) \in \N^N$} is characterized by the master equation +\be +\frac{dP(\xi,t)}{dt} = {\cal L^*} P(\xi,t) +\ee +where $P(\xi,t)$ represents the +probability mass function for the process $\Xi(t)$, that is +$$ +P(\xi,t) = Prob (\Xi(t) = \xi) +$$ +and +\begin{eqnarray} +{\cal L}^*P(\xi,t) +& = & +\sum_i {\cal L}^*_{i,i+1}P(\xi,t) \nonumber \\ +& = & +\sum_i 2\xi_i \left(1+ 2\xi_{i+1}\right) P(\xi^{i,i+1},t) ++ \left(1+2\xi_i\right)2\xi_{i+1} P(\xi^{i+1,i},t) \nonumber\\ +& & - 2\left(2\xi_i + \frac{1}{2}\right)\left(2\xi_{i+1} + \frac{1}{2}\right) P(\xi,t) ++ \frac{1}{2}P(\xi,t) +\end{eqnarray} +and $\xi^{i,j}$ denotes the configuration that is obtained by removing one particle +at $i$ and adding one particle at $j$. +\newpage +\item +\underline{In general, Duality means the following}: +there exists functions $O(x,\xi): \R^N \times \N^N \mapsto \R$ such that +the following equality between expectations for the two processes holds +\begin{center} +\fbox{\parbox{9cm}{ +\be +\E_x( O(X(t),\xi)) =\E_\xi(O(x,\Xi(t))) +\ee +}} +\end{center} +The subscripts in the expectations denote the initial conditions of the processes: +$X(0) =x$ on the left and $\Xi(0) = \xi$ on the right. +More explicitly we have: +\be +\int dy O(y,\xi) p(y,t; x,0) = \sum_{\eta} O(x,\eta) P(\eta,t; \xi,0) +\ee +To prove duality it is sufficient to show that +\be +\label{main} +L O(x,\xi) = {\cal L} O(x,\xi) +\ee +where $L$, that is working on $x$, is the adjoint of $L^*$ and ${\cal L}$, that is working on $\xi$, +is the adjoint of ${\cal L}^*$. +Indeed we have: +\begin{eqnarray} +\E_x( O(X(t),\xi)) +& = & +\int dy O(y,\xi) p(y,t; x,0) \\ +& = & +\sum_{\eta} \int dy O(y,\eta) p(y,t; x,0) \delta_{\eta,\xi} \\ +& = & +\sum_{\eta} \int dy O(y,\eta) e^{tL^*} \delta(y-x) \delta_{\eta,\xi} \\ +& = & +\sum_{\eta} \int dy e^{tL} O(y,\eta) \delta(y-x) \delta_{\eta,\xi} \\ +& = & +\sum_{\eta} \int dy e^{t{\cal L}} O(y,\eta) \delta(y-x) \delta_{\eta,\xi} \\ +& = & +\sum_{\eta} \int dy O(y,\eta) e^{t{\cal L}^*} \delta(y-x) \delta_{\eta,\xi} \\ +& = & +\sum_{\eta} \int dy O(y,\eta) P(\eta,t;\xi,0) \delta(y-x) \\ +& = & +\sum_{\eta} O(x,\eta) P(\eta,t;\xi,0) \\ +& = & +\E_\xi(O(x,\Xi(t))) +\end{eqnarray} +\newpage +\item +\underline{For the present case, the proper function to be considered are} +\be +\label{Oss} +O(x,\xi) = \prod_{i} \frac{x_i^{2\xi_i}}{(2\xi_i-1)!!} +\ee +Let us check Eq.(\ref{main}) on this choice. We have +\begin{eqnarray*} +&& +L_{i,i+1} O(x,\xi) += +\left(\prod_{k\not\in\{i,i+1\}} \frac{x_k^{2\xi_k}}{(2\xi_k -1)!!}\right) +\times +\\ +&&\left(2\xi_{i+1}(2\xi_{i+1}-1) \frac{x_i^{2\xi_i+2}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}-2}}{(2\xi_{i+1} -1)!!} +- 2\xi_{i}(2\xi_{i+1}+1) \frac{x_i^{2\xi_i}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}}}{(2\xi_{i+1} -1)!!} +\right. +\\ +&&\left.- 2\xi_{i+1}(2\xi_{i}+1) \frac{x_i^{2\xi_i}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}}}{(2\xi_{i+1} -1)!!} ++2\xi_{i}(2\xi_{i}-1) \frac{x_i^{2\xi_i-2}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}+2}}{(2\xi_{i+1} -1)!!} +\right) +\\ +\end{eqnarray*} +which implies +\begin{eqnarray*} +L_{i,i+1} O(x,\xi) +& = & +\Big(2\xi_{i+1}(2\xi_{i}+1) [O(x,\xi^{i+1,i})-O(x,\xi)] +\\ +&& +\;+\;2\xi_{i}(2\xi_{i+1}+1) [O(x,\xi^{i,i+1})-O(x,\xi)]\Big) +\\ +& = & +{\cal L}_{i,i+1} O(x,\xi) +\end{eqnarray*} + +\item \underline{How to find the proper normalization?} +Suppose that we are in the general following situation: +\begin{itemize} +\item We have a generator $L$ of a Markov process $X(t)$. +\item We know its stationary measure $\pi(x)$: +\be +L^* \pi(x) = 0 +\ee +\item We have functions $f(x,\xi)$ for which the following holds: +\be +\label{aaa} +L f(x,\xi) = \sum_{\eta} r(\xi,\eta) f(x,\eta) +\ee +with +\be +\label{bbb} +r(\xi,\eta) \ge 0 \qquad \mbox{if}\quad \xi \neq \eta +\ee +\be +\label{ccc} +r(\xi,\xi) \le 0 \qquad \mbox{if}\quad \xi = \eta +\ee +\end{itemize} +The matrix $r$ resembles the generator of a dual Markov process, +but it is not because it does not satisfy the condition +$\sum_{\eta} r(\xi,\eta) = 0$. +In order to find the generator of the dual process we proceed as +follows: +\begin{enumerate} +\item Define +\be +m(\xi) = \int f(x,\xi) \pi(x) dx +\ee +\item Define +\be +q(\xi,\eta)= m(\xi)^{-1} r(\xi,\eta) m(\eta) +\ee +\item Define +\be +O(x,\xi) = m(\xi)^{-1} f(x,\xi) +\ee +\end{enumerate} +Then the matrix $q$ can be seen as the generator of the dual Markov process $\Xi(t)$, that is +\be +L O(x,\xi) = \sum_{\eta} q(\xi,\eta) O(x,\eta) +\ee +with +\be +q(\xi,\eta) \ge 0 \qquad \mbox{if}\quad \xi \neq \eta +\ee +\be +q(\xi,\xi) \le 0 \qquad \mbox{if}\quad \xi = \eta +\ee +\be +\sum_{\eta} q(\xi,\eta) = 0 +\ee +Indeed we have: +\begin{eqnarray} +L O(x,\xi) +&=& +L m(\xi)^{-1} f(x,\xi) \nonumber \\ +&=& +m(\xi)^{-1} \sum_{\eta} r(\xi,\eta) f(x,\eta) \nonumber \\ +&=& +m(\xi)^{-1} \sum_{\eta} m(\xi)q(\xi,\eta) m(\eta)^{-1} m(\eta) O(x,\eta)\nonumber \\ +&=& +\sum_{\eta} q(\xi,\eta) O(x,\eta) +\end{eqnarray} +and +\begin{eqnarray} +\sum_{\eta} q(\xi,\eta) +&=& +\sum_{\eta} m(\xi)^{-1} r(\xi,\eta) m(\eta) \nonumber \\ +&=& +m(\xi)^{-1} \sum_{\eta} r(\xi,\eta) \int f(x,\eta) \pi(x) dx \nonumber \\ +&=& +m(\xi)^{-1} \int L f(x,\xi) \pi(x) dx \nonumber \\ +&=& +m(\xi)^{-1} \int f(x,\xi) L^* \pi(x) dx \nonumber \\ +&=& +0 +\end{eqnarray} + + + +\item \underline{Our case}. Among all the invariant measure +of the $X(t)$ process, namely the normalized function with +spherical symmetry $p(x) = p(\sum_i x_i^2)$, a special role is +played by the Gibbs measure +$$ +\pi(x) += \left(\frac{\beta}{2\pi}\right)^{(N/2)} e^{-\beta\sum_i \frac{x_i^2}{2}} += \left(\frac{\beta}{2\pi}\right)^{(N/2)} \prod_i e^{-\beta\frac{x_i^2}{2}} +$$ +which is selected as soon as the system is placed in contact with +thermal bath working at inverse temperature $\beta$. +Moreover: If $Z$ is a centered Gaussian, namely $Z\sim N(0,\sigma^2)$, +then +$$ +\E(Z^{2n}) = \sigma^{2n} (2n-1)!! +$$ +If one start from +$$ +f(x,\xi) = \prod_i x_i^{2\xi} +$$ +which satisfy (\ref{aaa}),(\ref{bbb}),(\ref{ccc}) and apply +the previous procedure, one arrives to (\ref{Oss}). + +{\bf Remark:} Note that, in applying the procedure, the +dependence on $\beta$ disappear!!!! +\end{itemize} + + +\section{Quantum language} + + +Here we start from a quantum spin chain +$$ +H = - 4 \sum_i \left( K^+_iK^-_{i+1} + K^-_iK^+_{i+1} -2 K^0_iK^0_{i+1} + \frac{1}{8}\right) +$$ +where the spin $K_i$'s satisfy the SU(1,1) algebra +\begin{eqnarray} +\label{commutatorsSU11} +[K_i^{0},K_i^{\pm}] &=& \pm K_i^{\pm} \nonumber \\ +{[}K_{i}^{-},K_{i}^{+}{]} &=& 2K_i^{0} +\end{eqnarray} +We are going to see the Schr\"odinger equation with imaginary time +\begin{equation} +\label{schroedinger} +\frac{d}{dt}|\psi(t) \rangle = -H |\psi(t)\rangle\;. +\end{equation} +as the evolution equation for the probability distribution of +a Markovian stochastic process. +\begin{itemize} +\item +\underline{The Hamiltonian possesses the SU(1,1) invariance}. If we define +\be +K^+ = \sum_{i} K_i^+ +\ee +\be +K^- = \sum_{i} K_i^- +\ee +\be +K^0 = \sum_{i} K_i^0 +\ee +we find that +\be +[H,K^+] = 0 +\ee +\be +[H,K^-] = 0 +\ee +\be +[H,K^0] = 0 +\ee +\item +\underline{Since $[H,K^+] = 0$} there exist a basis to study the stochastic process associated to +$H$ where \underline{$K^+$ is diagonal}. We might consider the following representation +\begin{eqnarray} +\label{Koper} +K^+_i &=& \frac{1}{2} x_{i}^2 \nonumber \\ +K^-_i &=& \frac{1}{2} \frac{\partial^2}{\partial x_{i}^2} \nonumber \\ +K^o_i &=& \frac{1}{4} \left\{\frac{\partial}{\partial x_{i}} x_{i} + + x_{i} \frac{\partial}{\partial x_{i}} \right \} +\end{eqnarray} +If we use this representation then +$$ +H = -L^* +$$ +and the probability density function for the $X(t)$ process is encoded in +the state $|\psi(t)\rangle$, namely +\begin{equation} +|\psi(t) \rangle = \int dx p(x,t) |x\rangle +\end{equation} +where we have introduced the notation $|x\rangle$ to denote a completely +localized state, that is a vector which together with its transposed +$\langle x|$ form a complete basis of a Hilbert space and its dual: +\begin{equation} +\langle x|x' \rangle = \delta(x-x') +\end{equation} +It immediately follows that +\begin{equation} +\langle x|\psi(t) \rangle = p(x,t) +\end{equation} +To compute expectation with respect to the $X(t)$ process +we introduce the flat state +\begin{equation} +\langle - | = \int dx \;\langle x| +\end{equation} +which is such that +\begin{equation} +\langle - | x\rangle = 1 +\end{equation} +Then for any observable $A = A(X(t))$ we have that its expectation value +at time $t$ can be written as +\begin{equation} +\langle A(t) \rangle_x = \int dy \,A(y)\, p(y,t;x,0) = \langle -|A| \psi(t) \rangle_x = \langle -|A e ^{-tH}| x\rangle +\end{equation} +\item +\underline{Since $[H,K^0] = 0$} there exist a basis to study the stochastic process associated to +$H$ where \underline{$K^0$ is diagonal}. We might consider the following representation +\begin{eqnarray} +\label{Koper2} +K^+_i|\xi\rangle &=& \left(\frac{1}{2} + \xi\right) |\xi+1\rangle\nonumber \\ +K^-_i|\xi\rangle &=& \xi |\xi-1\rangle\nonumber \\ +K^o_i|\xi\rangle &=& \left(\xi + \frac{1}{4}\right) |\xi\rangle +\end{eqnarray} +where $|\xi\rangle$ denotes a vector which together with its transposed +$\langle \xi|$ form a complete basis of a Hilbert space and its dual, that is +\begin{equation} +\langle \xi|\eta \rangle = \delta_{\xi,\eta} +\end{equation} +If we use this representation then +$$ +H = -{\cal L}^* +$$ +and the probability mass function for the $\Xi(t)$ process is encoded in +the state $|\phi(t)\rangle$, namely +\begin{equation} +|\phi(t) \rangle = \sum_{\xi} P(\xi,t) |\xi\rangle +\end{equation} +It immediately follows that +\begin{equation} +\langle \xi|\phi(t) \rangle = P(\xi,t) +\end{equation} +To compute expectation with respect to the $\Xi(t)$ process +we introduce the flat state +\begin{equation} +\langle -_{dual} | = \sum_{\xi} \;\langle \xi| +\end{equation} +which is such that +\begin{equation} +\langle -_{dual} | \xi\rangle = 1 +\end{equation} +Then for any observable $A=A(\Xi(t))$ we have that its expectation value +at time $t$ can be written as +\begin{equation} +\langle A(t) \rangle_\xi = \sum_{\eta}\,A(\eta)\, p(\eta,t;\xi,0) = \langle -_{dual}|A| \phi(t) \rangle_{\xi} = \langle -_{dual}|A e ^{-tH}| \xi\rangle +\end{equation} +\item +\underline{The claim is the following: Duality, in general, is going from the basis +where}\\ +\underline{one generator of the group is diagonal to a basis where another generator of}\\ +\underline{ the group is diagonal.} + +In our case we change from a basis where $K^+$ is diagonal to the base where $K^0$ is diagonal. + +\begin{eqnarray} +\langle - |\prod_i\frac{(2K_i^+)^{\xi_i}}{(2\xi_i-1)!!}|\psi(t)\rangle_x +& = & +\int dy \; \langle y |\prod_i\frac{(2K_i^+)^{\xi_i}}{(2\xi_i-1)!!} e^{tL^*}|x\rangle \nonumber \\ +& = & +\sum_{\eta} \int dy \; \langle y |\prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!}e^{tL^*}|x\rangle \langle \eta|\xi\rangle\nonumber \\ +& = & +\sum_{\eta} \int dy \; \langle y| \otimes \langle \eta| \prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!} e^{tL^*} | x\rangle \otimes|\xi\rangle\nonumber \\ +& = & +\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{tL} \prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta\rangle \nonumber \\ +& = & +\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{tL} \prod_i\frac{y^{2\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\ +& = & +\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{t{\cal L}} \prod_i\frac{y_i^{2\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\ +& = & +\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{t{\cal L}} \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\ +& = & +\sum_{\eta} \int dy \; \langle y| \otimes \langle \eta |\prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} e^{t{\cal L}^*} | x\rangle \otimes|\xi \rangle\nonumber \\ +& = & +\sum_{\eta} \int dy \; \langle \eta | \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} e^{t{\cal L}^*} |\xi \rangle \langle y | x\rangle \nonumber \\ +& = & +\sum_{\eta} \langle \eta | \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} |\phi(t)\rangle_{\xi} \nonumber \\ +& = & +\langle -_{dual} |\prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!}|\phi(t)\rangle_{\xi} +\end{eqnarray} + + +\end{itemize} + +\section{General k} + +A convenient $(2j+1)$-dimensional representation of the SU(2) algebra is given by +\begin{eqnarray} +J^+_i |n_i\rangle &=& (2j-n_i) |n_i+1\rangle \nonumber \\ +J^-_i |n_i\rangle &=& n_i |n_i-1\rangle \nonumber \\ +J^0_i |n_i\rangle &=& (n_i-j) |n_i\rangle +\end{eqnarray} +where the quantum numbers $n_i\in\{0,1,\ldots,2j\}$. +{\bf Note that in this representation the adjoint of $J^+_i$ is not +$J^-_i$, UNLESS $j=1/2$}. + +A matrix representation is: +$$ +J^+ = \left( +\begin{array}{cccc} + 0 & & & \\ + 2j & \ddots & & \\ + & \ddots & \ddots & \\ + & & 1 & 0\\ +\end{array}\right) +\qquad +J^- = \left( +\begin{array}{cccc} + 0 & 1 & & \\ + & \ddots & \ddots & \\ + & & \ddots & 2j \\ + & & & 0 \\ +\end{array}\right) +\qquad +J^0 = \left( +\begin{array}{cccc} + -j & & & \\ + & \ddots & & \\ + & & \ddots & \\ + & & & j\\ +\end{array}\right) +$$ + +In the SU(1,1) case one can use the infinite dimensional representation +\begin{eqnarray} +\label{newrepresentationsu11} +K^+_i |n_i\rangle &=& (2k+n_i) |n_i+1\rangle \nonumber \\ +K^-_i |n_i\rangle &=& n_i |n_i-1\rangle \nonumber \\ +K^0_i |n_i\rangle &=& (n_i+k) |n_i\rangle +\end{eqnarray} +where the quantum numbers $n_i\in\{0,1,2,\ldots\}$. +A matrix representation is: +$$ +K^+ = \left( +\begin{array}{cccc} + 0 & & & \\ + 2k & \ddots & & \\ + & 2k+1 & \ddots & \\ + & & \ddots & \ddots\\ +\end{array}\right) +\qquad +K^- = \left( +\begin{array}{cccc} + 0 & 1 & & \\ + & \ddots & 2 & \\ + & & \ddots & \ddots \\ + & & & \ddots \\ +\end{array}\right) +\qquad +K^0 = \left( +\begin{array}{cccc} + k & & & \\ + & k+1 & & \\ + & & k+2 & \\ + & & & \ddots\\ +\end{array}\right) +$$ +Let's check that in this representation the operator is stochastic. +I will do it for the bulk: +\begin{eqnarray} +L_{i,i+1}|n_i,n_{i+1}\rangle +&=& +(2k+n_i) n_{i+1}|n_i +1 ,n_{i+1}-1\rangle \nonumber\\ +&+& +n_i(2k+n_{i+1})|n_i -1 ,n_{i+1}+1\rangle \nonumber\\ +&+& +(-2(n_i+k)(n_{i+1}+k)+2k^2)|n_i,n_{i+1}\rangle +\end{eqnarray} +The sum of the rates is +$$ +(2k+n_i) n_{i+1}+ +n_i(2k+n_{i+1}) +-2(n_i+k)(n_{i+1}+k)+2k^2 =0 +$$ + + + + + + + + + +% \end{document} -- cgit v1.2.3-54-g00ecf From 57c597f34858eb256a1c681ab931cf0ec665cec3 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 13:49:20 +0100 Subject: Letter format and bibliography updates. --- bezout.bib | 17 +- bezout.tex | 2 +- cover.tex | 1419 ++++-------------------------------------------------------- 3 files changed, 87 insertions(+), 1351 deletions(-) diff --git a/bezout.bib b/bezout.bib index 8688ed2..9e600f0 100644 --- a/bezout.bib +++ b/bezout.bib @@ -267,6 +267,16 @@ doi = {10.1007/jhep11(2012)023} } +@book{Mezard_2009_Information, + author = {Mézard, Marc and Montanari, Andrea}, + title = {Information, physics, and computation}, + publisher = {Oxford University Press}, + year = {2009}, + address = {Great Clarendon Street, Oxford}, + isbn = {9780198570837}, + series = {Oxford Graduate Texts} +} + @article{Nguyen_2014_The, author = {Nguyen, Hoi H. and O'Rourke, Sean}, title = {The Elliptic Law}, @@ -294,12 +304,7 @@ url = {https://doi.org/10.2307%2F2371510}, doi = {10.2307/2371510} } -@book{mezard2009information, - title={Information, physics, and computation}, - author={Mezard, Marc and Montanari, Andrea}, - year={2009}, - publisher={Oxford University Press} -} + @inproceedings{Scorzato_2016_The, author = {Scorzato, Luigi}, title = {The {Lefschetz} thimble and the sign problem}, diff --git a/bezout.tex b/bezout.tex index 96ad257..d7f4dc7 100644 --- a/bezout.tex +++ b/bezout.tex @@ -44,7 +44,7 @@ Spin-glasses have long been considered the paradigm of many variable `complex landscapes,' a subject that includes neural networks and optimization problems, -most notably constraint satisfaction \cite{mezard2009information}. The most tractable family of these +most notably constraint satisfaction \cite{Mezard_2009_Information}. The most tractable family of these are the mean-field spherical $p$-spin models \cite{Crisanti_1992_The} (for a review see \cite{Castellani_2005_Spin-glass}) defined by the energy \begin{equation} \label{eq:bare.hamiltonian} diff --git a/cover.tex b/cover.tex index 586fa38..2d2fc10 100644 --- a/cover.tex +++ b/cover.tex @@ -1,1358 +1,89 @@ - -\documentclass[12pt,reqno,a4paper,twoside]{article} -% \ProvidesPackage{makra} -\usepackage{amsmath,amsthm,amstext,amscd,amssymb,euscript} -%,showkeys} -%,times} -\usepackage{epsf} -\usepackage{color} -\usepackage{verbatim} -\usepackage{graphicx} -\usepackage{esint} -\usepackage{tikz} -\usepackage{setspace} -\usepackage{mathrsfs} - -\usepackage{todonotes} - -%\usepackage{natbib} - - - -\usepackage{bm} -\usepackage[normalem]{ulem} - - -\textwidth 6in -\topmargin -0.50in -\oddsidemargin 0in -\evensidemargin 0in -\textheight 9.00in -%\pagestyle{plain} -%%%%%%%%%%%%%%%%%% Macros %%%%%%%%%%% -\def\mybox #1{\fbox{\parbox{5.8in}{#1}}} -\newcommand{\m}[1]{{\marginpar{\scriptsize #1}}} - -\def\mep{\mathbf{mep}_{n}^{\delta}} -\def\r{{\mathbf r}} -\def\O{{\mathcal{O}}} - -\def\I{{\mathcal{I}}} -\def\fee{\mathcal{F}} - -\def\F{{\EuScript{F}}} - -\renewcommand{\phi}{\varphi} -\newcommand{\compose}{\circ} -\renewcommand{\subset}{\subseteq} -\renewcommand{\emptyset}{\varnothing} -\newcommand{\interval}{[\underline\alpha,\overline \alpha]} -\def\liminfn{\liminf_{n\to\infty}} -\def\limsupn{\limsup_{n\to\infty}} -\def\limn{\lim_{n\to\infty}} -\def\disagree{\not\longleftrightarrow} -\newcommand{\Zd}{\mathbb Z^d} -\newcommand{\kk}{\mathbf k} -\renewcommand{\Pr}{\mathbb P} -\newcommand{\dist}{\text{dist}} -\newcommand{\Cal}{\mathcal} -\def\1{ {\mathit{1} \!\!\>\!\! 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+\usepackage[T1]{fontenc} % vector fonts plz +\usepackage{newtxtext,newtxmath} % Times for PR +\usepackage[ + colorlinks=true, + urlcolor=purple, + citecolor=purple, + filecolor=purple, + linkcolor=purple +]{hyperref} % ref and cite links with pretty colors +\usepackage{xcolor} + +\signature{ + Jaron Kent-Dobias \& Jorge Kurchan } -% -\newcommand{\graphh}{ -\begin{tikzpicture}[scale=.7] -\draw[gray, thick] (1,-1.5)--(0,0) -- (2,3)--(4,3.4)--(5.0,2)--(4.9,1)--(3.0,-1.2)--(1,-1.5); -\tikzstyle myBG=[line width=3.5pt,opacity=1.0] -% -\draw[white,myBG] (3.2,1.5) -- (5.2,3); -\draw[gray, thick] (3.2,1.5) -- (5.2,3); -% -\draw[gray, thick] (0,0)--(-1.5,0); -\draw[gray, thick] (3,-1.2)--(4,-2); -\draw[gray, thick] (4,3.4)--(4.3,4.6); -\draw[gray, thick] (1,-1.5)--(2,0.2)--(0,0); -\draw[gray, thick] (2,0.2)--(4.9,1); -\draw[gray, thick] (3,-1.2)--(2,0.2)--(3.2,1.5)--(2,3); -\draw[gray, thick] (3.2,1.5)--(4,3.4); -\draw[gray, thick] (3.2,1.5)--(5,2); -\draw[gray, thick] (4,3.4)--(5.2,3)--(5.7,2.2)--(5.7,1.3)--(4.9,1); -\draw[gray, thick] (5.7,2.2)--(5,2); -% -% -% -\filldraw[black!70] (0,0) circle (2pt); -\filldraw[black!70] (1,-1.5) circle (2pt); -\filldraw[black!70] (2,3) circle (2pt); -\filldraw[black!70] (4,3.4) circle (2pt); -\filldraw[black!70] (5,2) circle (2pt); -\filldraw[black!70] (4.9,1) circle (2pt); -\filldraw[black!70] (2,0.2) circle (2pt); -\filldraw[black!70] (3.2,1.5) circle (2pt); -% \filldraw[black!70] (5.7,1.3) circle (2pt); -\filldraw[black!70] (5.7,2.2) circle (2pt); -\filldraw[black!70] (5.2,3) circle (2pt); -\filldraw[black!70] (3,-1.2) circle (2pt); -% -% -\filldraw[black!70] (-1.5,0) circle (2pt) node[anchor=east] {$\Gamma_3,T$}; -\filldraw[white] (-1.5,0) circle (1pt); -\filldraw[black!70] (4,-2) circle (2pt) node[anchor=west] {$\Gamma_2,T$}; -\filldraw[white] (4,-2) circle (1pt); -\filldraw[black!70] (4.3,4.6) circle (2pt) node[anchor=west] {$\Gamma_1,T$}; -\filldraw[white] (4.3,4.6) circle (1pt) ; -\end{tikzpicture} +\address{ + Laboratoire de Physique \\ + Ecole Normale Sup\'erieure \\ + 24, rue Lhomond \\ + 75005 Paris } -\setstretch{1.24} - \begin{document} - -\title{{\bf COVER LETTER \\`Complex complex landscapes'}} -%\footnote{{\bf Key-words}: } -%}} - -\author{ -Jaron Kent-Dobias - and -Jorge Kurchan +\begin{letter}{ + Editorial Office\\ + Physical Review Letters\\ + 1 Research Road\\ + Ridge, NY 11961 } -\maketitle - - - -\vspace{1.cm} - - -The subject of `Complex Landscapes', which started in the spin-glass literature, is concerned with functions (landscapes) of many variables, having a multiplicity of minimums, which are the objects of interest. Apart from its obvious interest for glassy systems, it has found a myriad applications in many domains: Computer Science, Ecology, Economics, Biology \cite{mezard2009information}. - -In the last few years, a renewed interest has developed for landscapes for which the variables are complex. There are a few reasons for this: {\em i)} in Computational Physics, there is the main obstacle of the `sign problem', and a strategy has emerged to attack it deforming the sampling space into complex variables. This is a most natural and promising path, and any progress made will have game-changing impact in solid state physics and lattice-QCD \cite{Cristoforetti_2012_New,Scorzato_2016_The}. -{\em ii)} At a more basic level, following the seminal work of E. Witten \cite{Witten_2010_A,Witten_2011_Analytic}, there has been a flurry of activity concerning the very definition of quantum mechanics, which requires also that one move into the complex plane. - -In all these cases, just like in the real case, one needs to know the structure of the `landscape', where are the saddle points and how they are connected, typical questions of `complexity'. -However, to the best of our knowledge, there are no studies extending the methods of the theory of -complexity to -complex variables. -We believe our paper will open a field that may find -numerous applications and will widen our theoretical view of complexity in general. - - +\opening{} + +The subject of `Complex Landscapes,' which started in the spin-glass +literature, is concerned with functions (landscapes) of many variables, having +a multiplicity of minimums, which are the objects of interest. Apart from its +obvious interest for glassy systems, it has found a myriad applications in +many domains: Computer Science, Ecology, Economics, Biology +\cite{Mezard_2009_Information}. + +In the last few years, a renewed interest has developed for landscapes for +which the variables are complex. There are a few reasons for this: {\em i)} in +Computational Physics, there is the main obstacle of the `sign problem', and a +strategy has emerged to attack it deforming the sampling space into complex +variables. This is a most natural and promising path, and any progress made +will have game-changing impact in solid state physics and lattice-QCD +\cite{Cristoforetti_2012_New,Scorzato_2016_The}. {\em ii)} At a more basic +level, following the seminal work of E. Witten +\cite{Witten_2010_A,Witten_2011_Analytic}, there has been a flurry of activity +concerning the very definition of quantum mechanics, which requires also that +one move into the complex plane. + +In all these cases, just like in the real case, one needs to know the structure +of the `landscape', where are the saddle points and how they are connected, +typical questions of `complexity'. However, to the best of our knowledge, +there are no studies extending the methods of the theory of complexity to +complex variables. We believe our paper will open a field that may find +numerous applications and will widen our theoretical view of complexity in +general. + +\closing{Sincerely,} +\end{letter} + +\bibliographystyle{unsrt} \bibliography{bezout} - \end{document} - - - - - - - - - - - - - - - -\section{The Kipnis-Marchioro-Presutti model} - -Consider the following process: -\begin{itemize} -\item -choose a pair of neighbouring sites and completely -exchange energy between them -\item -if the site is one of the borders, exchange completely energy with the bath. -\end{itemize} -each choice with probability $1/(N+1)$. From here onwards, we shall denote -$\tau$ a large time, sufficient for any two-site thermalisation. - -The evolution operator in one step is: -\begin{eqnarray} -U &=& \frac{1}{N+1} \left[ e^{-\tau L_1^*} + e^{-\tau L_N^*} + \sum_{i=1}^{N-1} e^{-\tau L^*_{i,i+1}} \right] -\nonumber \\ -&=& \frac{1}{N+1} \left[ e^{-2\tau (T_1 K^-_1 + K^o_1 + k) } + e^{-2\tau(T_L K^-_L + K^o_L +k) } - + \sum_{i=1}^{N-1} e^{ \frac{-\tau}{k} -(K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1} -+ 2k^2 )} \right] \nonumber \\ -~ -\end{eqnarray} -and the dynamics after $n$ steps is given by $U^n$. -Because we are considering large $\tau$, the terms in the sums are in fact projectors -onto the lowest eigenvalues of the exponents. We shall however keep the notation as it is -in order to stress the symmetry of the bulk terms. - -Let us now show that - at the level of energies - this dynamics yields the KMP process -{\em for $k=\frac{1}{2}$, that is $m=2$}. -Consider first a general $m$, and two neighbouring sites of coordinates $x = \{x_\alpha\}_{\alpha=1,\ldots,m}$, -$y=\{y_\alpha\}_{\alpha=1,\ldots,m}$. -If they are completely thermalised, it means that (cfr (\ref{bb}): -the joint probability density satisfies -\begin{equation} -\left(x_{\alpha} -\frac{\partial}{\partial y_{\beta}} - -y_{\beta}\frac{\partial}{\partial x_{\alpha}} - \right) p(x,y)=0 -\end{equation} -It is easy to see that this may happen if and only if -\begin{equation} -p(x,y)= p[ \sum_\alpha (x_\alpha^2+y_\alpha^2)] -\end{equation} -In particular let us consider the microcanonical measure -\begin{equation} -p(x,y)= \delta[ \sum_\alpha (x_\alpha^2+y_\alpha^2)-\epsilon ] -\end{equation} -Defining new random variables $\epsilon_1$ and $\epsilon_2$ -as the energies of the neighboring sites -\be -\epsilon_1 = \sum_\alpha x_\alpha^2 -\ee -\be -\epsilon_2 = \sum_\alpha y_\alpha^2 -\ee -then their joint probability density will be -\begin{equation} -p(\epsilon_1,\epsilon_2) = \frac{S_m^2}{4} \delta(\epsilon_1+\epsilon_2-\epsilon) -\epsilon_1^{\frac{1}{2}-1} \epsilon_2^{\frac{1}{2}-1} -\end{equation} -where $S_m$ denotes the surface of the unit sphere in $m$ dimension -\be -S_m = \frac{m \pi^{m/2}}{\Gamma(\frac{1}{2}+1)} -\ee -{\em This yields a flat distribution for $m=2$, i.e. the KMP model.} - - - - -\section{Dual model} - - -The expectation value of an observable at time $t$, starting from an initial -distribution $|init\rangle$ is: - - -\begin{equation} - = \langle - | O e^{-Ht} | init \rangle -\end{equation} -where $\langle - |$ is a constant. -Taking the adjoint $ x_i \to x_i$, $\partial_i \to -\partial_i$: -\begin{equation} - = \langle - | O e^{-Ht} | init \rangle= \langle init| e^{-H^\dag t} O |- \rangle -\end{equation} -where $H^\dag(K^\pm, K^o)=H( K^\pm, -K^o)$ (because of the change of signs of the derivatives) -\begin{eqnarray} --H^\dag&=& \frac{4}{1} \sum_i \left( -K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1} -+ \frac{m^2}{8} \right) -\nonumber\\ -&+&2 \left(T_1 K^-_1 + K^o_1 + \frac{1}{4}\right) -+2 \left(T_L K^-_L + K^o_L +\frac{1}{4}\right) -\end{eqnarray} -In particular, for the generating function we had chosen - \begin{equation} - O |- \rangle = \Pi_i \frac{x_i^{2 \xi_i}}{(2\xi_i -1)!!}|-\rangle=|\xi_1,...,\xi_N\rangle -\end{equation} - -Considered as an operator acting on `particle number', as counted by $K^o$, $H^\dag$ does not -conserve the probability. -The trick we used can be expressed as follows: introduce the particle number $\xi_o$ and $\xi_{N+1}$ -and the operators $A^+_o$ and $A^+_{N+1}$, which create particles in boundary sites with unit rate. -We consider now the {\em enlarged} process generated by -\begin{eqnarray} --H^{dual}&=& \frac{4}{1} \sum_i \left( -K^+_i K^-_{i+1} + K^-_i K^+_{i+1} - 2 K^o_i K^o_{i+1} -+ \frac{m^2}{8} \right) -\nonumber\\ -&+&2 \left(A^+_o K^-_1 + K^o_1 - \frac{1}{4}\right) -+2 \left( A^+_{N+1} K^-_N + K^o_N -\frac{1}{4}\right) -\end{eqnarray} -which conserves ({\it seems}) particle number and probability. -We wish to prove that: - -\begin{eqnarray} - &=& \langle init| e^{-H^\dag t} |\xi_1,...,\xi_N \rangle \nonumber \\ -&=& \sum_{\xi_o,\xi_{N+1}} - T_1^{\xi_o} T_{L}^{\xi_{N+1}} \langle \xi_o \xi_{N+1} | \otimes \langle - init| e^{-H^{dual} t} |\xi_1,...,\xi_N \rangle \otimes - |\xi_o=0,\xi_{N+1}=0 \rangle \nonumber \\ -\label{ggg} -\end{eqnarray} - - -I think the proof is obvious, because developing the exponential of $H^{dual}$ all the $A^+$ can be -collected because they commute with everything else, and the experctation value -\begin{equation} -\sum_{\xi_o} T_1^{\xi_o} \langle \xi_o |[A^+_o]^r |\xi_o=0 \rangle = T_1^r - \end{equation} -just puts back as many $T$'s as necessary. - -I do not know exactly how to use (\ref{ggg}) in general, but in the large time limit the evolution -voids the chain of particles - - -\section{Dual of KMP} - -I think that the argument runs through without changes if we use $U$ defined for the KMP model. -We just have to note that each term corresponds to an evolution of two sites (or a site and the bath) -and so in the dual it corresponds to sharing the particles between those two sites, or emptying -the sites at the borders. - -{\bf: NOTE by Cristian} - -We can check that the duality function chosen in the original paper by KMP -do coincide with the duality function of our process for $m=2$ (and the random -variables are the energies). -Indeed we start from -\be -f(x,\xi) = \prod_i (\sum_{\alpha} x_{i,\alpha}^2)^{\xi} -\ee -When the bath have equal temperature (let's us choose T=1) then the stationary -measure is -\be -\pi(x) = \prod_i \frac{1}{(2\pi)^{m/2}} \exp\left(-\sum_{\alpha}\frac{x_{i,\alpha}^2}{2}\right) -\ee -Let us focus on a fixed $i$ (that is in this short computation we write $x$ for $x_i$). -We have -\begin{eqnarray} -\E(f(x,\xi)) -&=& -\int dx_1 \cdots \int dx_m (x_1^2+\ldots + x_m^2)^{\xi} \exp-\left(\frac{x_{1}^2}{2}+\ldots+\frac{x_{1}^2}{2}\right) -\nonumber \\ -& = & -\int dr S_m r^{2\xi} \exp-\left(\frac{r^2}{2}\right) -\nonumber \\ -& = & -\frac{\frac{1}{2}\Gamma(\frac{1}{2}+\xi)}{\Gamma(\frac{m}{2}+1)} 2^\xi -\nonumber \\ -\end{eqnarray} -Special cases: -\begin{itemize} -\item $m=1$ - -$$ -\E(f(x,\xi)) = (2\xi-1)!! -$$ -where one uses that $\Gamma(\frac{1}{2}+\xi)= \frac{\sqrt{\pi}(2\xi-1)!!}{2^{\xi}}$ and $\Gamma(\frac{3}{2}) = \frac{\sqrt{\pi}}{2}$ -\item $m=2$ - -$$ -\E(f(x,\xi)) = \xi! 2^\xi -$$ -where one uses that $\Gamma(1+\xi)= \xi!$ and $\Gamma(2) = 1$. -Thus, if one defines the energies as -$$ -\epsilon_i = \sum_{\alpha}\frac{x_{i,\alpha}^2}{2} -$$ -one recover the choice of KMP for the dual function -$$ -O(\epsilon_i,\xi) = \prod_i \frac{\epsilon_i^{\xi_i}}{\xi_i!} -$$ -\end{itemize} - - - - - - -\section{ Dual of SEP: here goes an outline of how to proceed for the SSEP} - - -\be -H=-L_{SEP}^* -\ee -\begin{eqnarray} -L^*_{SEP} &=& \frac{1}{j} - \sum_i \left(J^+_i J^-_{i+1} + J^-_i J^+_{i+1} + 2 J^o_i J^o_{i+1} - - 2 j^2 \right)\\ -&+&\alpha (J^-_1 - J^o_1-j) + \gamma (J^+_1 + J^o_1-j) -+ \delta (J^-_L - J^o_L-j) + \beta (J^+_L + J^o_L-j)\nonumber -\end{eqnarray} -The factor $1/j$ is analogous to the factor $1/m$ in (\ref{bb}). -The operators $J^+_i, J^-_i, J^o_i$ act on the Hilbert space - corresponding to $0 \le r \le n$ particles per site $\otimes_i |r\rangle_i$ -as follows: -\begin{eqnarray} -J^+_i |r\rangle_i &=& (2j-r) |r+1\rangle_i \nonumber \\ - J^-_i |r\rangle_i &=& r |r-1\rangle_i \nonumber \\ -J^o_i |r\rangle_i &=& (r-j) |r\rangle_i -\end{eqnarray} - -The conjugation properies are as follows. There is an operator $Q$, -{\em diagonal in this basis } (I give the expression below), such that: -\begin{equation} -[J^+_i]^\dag = Q[J^-_i]Q^{-1} \qquad [J^-_i]^\dag = Q[J^+_i]Q^{-1} -\end{equation} -while $[J^z_i]^\dag=J^z_i= Q[J^z_i]Q^{-1}$. - - - -The expectation value of an observable at time $t$, starting from an initial -distribution $|init\rangle$ is: - - -\begin{equation} - = \langle - | O e^{-Ht} | init \rangle -\end{equation} -where $\langle - |$ is a constant. -As before: -\begin{eqnarray} - &=& \langle - | O e^{-Ht} | init \rangle= -\langle init| e^{-H^\dag t} O |- \rangle= \nonumber \\ -& & \langle init|Q e^{-{\bar H} t} Q^{-1}O |- \rangle= -\langle init|Q \; e^{-{\bar H} t} Q^{-1}O Q Q^{-1} |- \rangle -\end{eqnarray} - - -{\em $ {\bar H}$ is the same operator as $H$ but with -$J^+$ substituted by $J^-$, and vice-versa.} -Our job is now to make the rotation that will eliminate the $J^+$'s in -the border terms of $ {\bar H}$. - - - - -The transformation is of the form -\begin{eqnarray} -e^{\mu J^+} J^+ e^{-\mu J^+}&=&J^+ \nonumber \\ -e^{\mu J^+} J^o e^{-\mu J^+} &=&J^o - \mu J^+ \nonumber \\ -e^{\mu J^+} J^- e^{-\mu J^+} &=& J^- + 2 \mu J^o - \mu^2 J^+ -\end{eqnarray} -for suitable $\mu$. -Putting $\mu=-1$, we get that {\bf the bulk term is left invariant, -precisely because of the SU(2) symmetry}. The boundary terms {\bf of $\bar H$} -transform further into: -\begin{eqnarray} -& e^{\mu J^+_1} \left[ \alpha (J^+_1 - J^o_1-j) + \gamma (J^-_1 + J^o_1-j) -\right] e^{-\mu J^+_1}= \nonumber \\ & \gamma(J^-_1 + 2 \mu J^o_1 - \mu^2 -J^+_1 +J^o_1 - \mu J^+_1 -j) + \alpha (J^+_1 - J^o_1 + \mu J^+_1 -j) -= \nonumber \\ -& \alpha(- J^o_1 -j) + \gamma (J^-_1 -J^o_1 -j) -\label{trans} -\end{eqnarray} -which is of the same form we have in the $SU(1,1)$ model. -The same can be done in the other boundary term. - -We thus get: -\begin{eqnarray} - &=& \langle - | O e^{-Ht} | init \rangle= -\langle init|Q \; e^{-{\bar H} t} Q^{-1}O Q Q^{-1} |- \rangle \nonumber\\ -&= & \langle init|Q e^{ \sum_i J^+_i} e^{-{\bar H_{dual}} t} - e^{ -\sum_i J^+_i} Q^{-1}O Q Q^{-1} |- \rangle \nonumber \\ -&= & \langle init|Q e^{ \sum_i J^+_i} e^{-{\bar H_{dual}} t} - e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i J^+_i} - e^{ -\sum_i J^+_i} |- \rangle \nonumber \\ - &= & \langle init|Q Q^{-1} e^{ \sum_i J^+_i} e^{-{\bar H_{dual}} t} - e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i J^+_i} |-_{dual} \rangle -\end{eqnarray} -where we have defined $H_{dual}$ as the transformed Hamiltonian. - -We now have to study $ |-_{dual} \rangle \equiv e^{ -\sum_i J^+_i} - Q^{-1} |- \rangle$ -Because we know that terms like those proportional to $\gamma$ and $\alpha$ -anihilate the measure to the left: -\begin{eqnarray} -& & \langle - | (J^-_i - J^o_i-j) =0\nonumber \\ -& & \langle - | (J^+_i + J^o_i-j) =0 -\end{eqnarray} -this implies that in the new variables and following all the transformations -(cfr (\ref{trans})): -\begin{eqnarray} -& & (J^-_i -J^o_i -j)e^{ -\sum_i J^+_i} Q^{-1} |- \rangle= 0 \nonumber \\ -& & ( -J^o_i -j)e^{ -\sum_i J^+_i} Q^{-1} |- \rangle =0 -\end{eqnarray} -which implies that $( J^o_i +j) |-_{dual} \rangle= J^-_i |-_{dual} \rangle=0$, -and this means that -\begin{equation} -J^o_i |-_{dual} \rangle =-j |-_{dual} \rangle -\end{equation} -is the vacuum of particles in this base! - -All in all we are left with: -\begin{eqnarray} - &=& \langle init|Q \; e^{ \sum_i J^+_i} - e^{-{\bar H_{dual}} t} e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i J^+_i} - |-_{dual} \rangle \nonumber \\ - &=& \langle init|Q \; e^{ \sum_i J^+_i} - e^{-{\bar H_{dual}} t} {\hat O} - |-_{dual} \rangle -\end{eqnarray} -where $ {\hat O} \equiv e^{ -\sum_i J^+_i} Q^{-1}O Q e^{ \sum_i -J^+_i}$. We have to start with the -vacuum $ |-_{dual} \rangle$, then apply $ {\hat O} $, (which creates -particles because it contains many $J^+$'s), and then there is the -dual evolution. The final configuration has to be overlapped with -$\langle f| \equiv \langle init|Q \; e^{ \sum_i J^+_i}$. -For large times, there will be no particle left except in the two extra sites -in the borders. - -\section{Constructive approach} - -Here I would like to say the following: if I have a modle of transport -of which I do not know if it has a Dual one, I can proceed as follows. -I take a small version with no baths and a few sites. I write the -evolution operator and I diagonalise it numerically. If there is a -non abelian group, the eigenvalues will be in degenerate -multiplets. Hence, if I find multiplets, then very probably there is a -dual model, if I do not, then there cannot be one. It would be nice -to show it with the KMP model with two or three sites. - -Another thing is to consider higher groups. $SU(3)$ has already been studied -for two kinds of particles. We know how to map to a dual in that -case, if it has not been done yet. - -\newpage -{\bf THIS PART HAS BEEN WRITTEN BY CRISTIAN} - -The aim of this file is to set notation in the two languages. -Let us focus on duality for the case we already know: -SU(1,1) model with $k=1/4$. To fix ideas let us consider only -the bulk part of the system with periodic boundary conditions. - -\section{Probabilistic language} -We have two stochastic Markovian process with continuous time. -\begin{itemize} -\item -\underline{The first process $X(t) \in \R^N$} is given by the Fokker-Planck equation: -\be -\frac{dp(x,t)}{dt} = L^* p(x,t) -\ee -where $p(x,t)$ represents the probability density -for the process $X(t)$, that is -$$ -p(x,t)dx = Prob (X(t)\in (x,x+dx)) -$$ -and -\begin{eqnarray} -L^*p(x,t) -& = & -\sum_i L^*_{i,i+1} p(x,t) \noindent\\ -& = & -\sum_i \left(x_i\frac{\partial}{\partial x_{i+1}} -x_{i+1}\frac{\partial}{\partial x_{i}}\right)^2 p(x,t) -\end{eqnarray} -\item -\underline{The second process $\Xi(t) \in \N^N$} is characterized by the master equation -\be -\frac{dP(\xi,t)}{dt} = {\cal L^*} P(\xi,t) -\ee -where $P(\xi,t)$ represents the -probability mass function for the process $\Xi(t)$, that is -$$ -P(\xi,t) = Prob (\Xi(t) = \xi) -$$ -and -\begin{eqnarray} -{\cal L}^*P(\xi,t) -& = & -\sum_i {\cal L}^*_{i,i+1}P(\xi,t) \nonumber \\ -& = & -\sum_i 2\xi_i \left(1+ 2\xi_{i+1}\right) P(\xi^{i,i+1},t) -+ \left(1+2\xi_i\right)2\xi_{i+1} P(\xi^{i+1,i},t) \nonumber\\ -& & - 2\left(2\xi_i + \frac{1}{2}\right)\left(2\xi_{i+1} + \frac{1}{2}\right) P(\xi,t) -+ \frac{1}{2}P(\xi,t) -\end{eqnarray} -and $\xi^{i,j}$ denotes the configuration that is obtained by removing one particle -at $i$ and adding one particle at $j$. -\newpage -\item -\underline{In general, Duality means the following}: -there exists functions $O(x,\xi): \R^N \times \N^N \mapsto \R$ such that -the following equality between expectations for the two processes holds -\begin{center} -\fbox{\parbox{9cm}{ -\be -\E_x( O(X(t),\xi)) =\E_\xi(O(x,\Xi(t))) -\ee -}} -\end{center} -The subscripts in the expectations denote the initial conditions of the processes: -$X(0) =x$ on the left and $\Xi(0) = \xi$ on the right. -More explicitly we have: -\be -\int dy O(y,\xi) p(y,t; x,0) = \sum_{\eta} O(x,\eta) P(\eta,t; \xi,0) -\ee -To prove duality it is sufficient to show that -\be -\label{main} -L O(x,\xi) = {\cal L} O(x,\xi) -\ee -where $L$, that is working on $x$, is the adjoint of $L^*$ and ${\cal L}$, that is working on $\xi$, -is the adjoint of ${\cal L}^*$. -Indeed we have: -\begin{eqnarray} -\E_x( O(X(t),\xi)) -& = & -\int dy O(y,\xi) p(y,t; x,0) \\ -& = & -\sum_{\eta} \int dy O(y,\eta) p(y,t; x,0) \delta_{\eta,\xi} \\ -& = & -\sum_{\eta} \int dy O(y,\eta) e^{tL^*} \delta(y-x) \delta_{\eta,\xi} \\ -& = & -\sum_{\eta} \int dy e^{tL} O(y,\eta) \delta(y-x) \delta_{\eta,\xi} \\ -& = & -\sum_{\eta} \int dy e^{t{\cal L}} O(y,\eta) \delta(y-x) \delta_{\eta,\xi} \\ -& = & -\sum_{\eta} \int dy O(y,\eta) e^{t{\cal L}^*} \delta(y-x) \delta_{\eta,\xi} \\ -& = & -\sum_{\eta} \int dy O(y,\eta) P(\eta,t;\xi,0) \delta(y-x) \\ -& = & -\sum_{\eta} O(x,\eta) P(\eta,t;\xi,0) \\ -& = & -\E_\xi(O(x,\Xi(t))) -\end{eqnarray} -\newpage -\item -\underline{For the present case, the proper function to be considered are} -\be -\label{Oss} -O(x,\xi) = \prod_{i} \frac{x_i^{2\xi_i}}{(2\xi_i-1)!!} -\ee -Let us check Eq.(\ref{main}) on this choice. We have -\begin{eqnarray*} -&& -L_{i,i+1} O(x,\xi) -= -\left(\prod_{k\not\in\{i,i+1\}} \frac{x_k^{2\xi_k}}{(2\xi_k -1)!!}\right) -\times -\\ -&&\left(2\xi_{i+1}(2\xi_{i+1}-1) \frac{x_i^{2\xi_i+2}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}-2}}{(2\xi_{i+1} -1)!!} -- 2\xi_{i}(2\xi_{i+1}+1) \frac{x_i^{2\xi_i}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}}}{(2\xi_{i+1} -1)!!} -\right. -\\ -&&\left.- 2\xi_{i+1}(2\xi_{i}+1) \frac{x_i^{2\xi_i}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}}}{(2\xi_{i+1} -1)!!} -+2\xi_{i}(2\xi_{i}-1) \frac{x_i^{2\xi_i-2}}{(2\xi_i -1)!!}\frac{x_{i+1}^{2\xi_{i+1}+2}}{(2\xi_{i+1} -1)!!} -\right) -\\ -\end{eqnarray*} -which implies -\begin{eqnarray*} -L_{i,i+1} O(x,\xi) -& = & -\Big(2\xi_{i+1}(2\xi_{i}+1) [O(x,\xi^{i+1,i})-O(x,\xi)] -\\ -&& -\;+\;2\xi_{i}(2\xi_{i+1}+1) [O(x,\xi^{i,i+1})-O(x,\xi)]\Big) -\\ -& = & -{\cal L}_{i,i+1} O(x,\xi) -\end{eqnarray*} - -\item \underline{How to find the proper normalization?} -Suppose that we are in the general following situation: -\begin{itemize} -\item We have a generator $L$ of a Markov process $X(t)$. -\item We know its stationary measure $\pi(x)$: -\be -L^* \pi(x) = 0 -\ee -\item We have functions $f(x,\xi)$ for which the following holds: -\be -\label{aaa} -L f(x,\xi) = \sum_{\eta} r(\xi,\eta) f(x,\eta) -\ee -with -\be -\label{bbb} -r(\xi,\eta) \ge 0 \qquad \mbox{if}\quad \xi \neq \eta -\ee -\be -\label{ccc} -r(\xi,\xi) \le 0 \qquad \mbox{if}\quad \xi = \eta -\ee -\end{itemize} -The matrix $r$ resembles the generator of a dual Markov process, -but it is not because it does not satisfy the condition -$\sum_{\eta} r(\xi,\eta) = 0$. -In order to find the generator of the dual process we proceed as -follows: -\begin{enumerate} -\item Define -\be -m(\xi) = \int f(x,\xi) \pi(x) dx -\ee -\item Define -\be -q(\xi,\eta)= m(\xi)^{-1} r(\xi,\eta) m(\eta) -\ee -\item Define -\be -O(x,\xi) = m(\xi)^{-1} f(x,\xi) -\ee -\end{enumerate} -Then the matrix $q$ can be seen as the generator of the dual Markov process $\Xi(t)$, that is -\be -L O(x,\xi) = \sum_{\eta} q(\xi,\eta) O(x,\eta) -\ee -with -\be -q(\xi,\eta) \ge 0 \qquad \mbox{if}\quad \xi \neq \eta -\ee -\be -q(\xi,\xi) \le 0 \qquad \mbox{if}\quad \xi = \eta -\ee -\be -\sum_{\eta} q(\xi,\eta) = 0 -\ee -Indeed we have: -\begin{eqnarray} -L O(x,\xi) -&=& -L m(\xi)^{-1} f(x,\xi) \nonumber \\ -&=& -m(\xi)^{-1} \sum_{\eta} r(\xi,\eta) f(x,\eta) \nonumber \\ -&=& -m(\xi)^{-1} \sum_{\eta} m(\xi)q(\xi,\eta) m(\eta)^{-1} m(\eta) O(x,\eta)\nonumber \\ -&=& -\sum_{\eta} q(\xi,\eta) O(x,\eta) -\end{eqnarray} -and -\begin{eqnarray} -\sum_{\eta} q(\xi,\eta) -&=& -\sum_{\eta} m(\xi)^{-1} r(\xi,\eta) m(\eta) \nonumber \\ -&=& -m(\xi)^{-1} \sum_{\eta} r(\xi,\eta) \int f(x,\eta) \pi(x) dx \nonumber \\ -&=& -m(\xi)^{-1} \int L f(x,\xi) \pi(x) dx \nonumber \\ -&=& -m(\xi)^{-1} \int f(x,\xi) L^* \pi(x) dx \nonumber \\ -&=& -0 -\end{eqnarray} - - - -\item \underline{Our case}. Among all the invariant measure -of the $X(t)$ process, namely the normalized function with -spherical symmetry $p(x) = p(\sum_i x_i^2)$, a special role is -played by the Gibbs measure -$$ -\pi(x) -= \left(\frac{\beta}{2\pi}\right)^{(N/2)} e^{-\beta\sum_i \frac{x_i^2}{2}} -= \left(\frac{\beta}{2\pi}\right)^{(N/2)} \prod_i e^{-\beta\frac{x_i^2}{2}} -$$ -which is selected as soon as the system is placed in contact with -thermal bath working at inverse temperature $\beta$. -Moreover: If $Z$ is a centered Gaussian, namely $Z\sim N(0,\sigma^2)$, -then -$$ -\E(Z^{2n}) = \sigma^{2n} (2n-1)!! -$$ -If one start from -$$ -f(x,\xi) = \prod_i x_i^{2\xi} -$$ -which satisfy (\ref{aaa}),(\ref{bbb}),(\ref{ccc}) and apply -the previous procedure, one arrives to (\ref{Oss}). - -{\bf Remark:} Note that, in applying the procedure, the -dependence on $\beta$ disappear!!!! -\end{itemize} - - -\section{Quantum language} - - -Here we start from a quantum spin chain -$$ -H = - 4 \sum_i \left( K^+_iK^-_{i+1} + K^-_iK^+_{i+1} -2 K^0_iK^0_{i+1} + \frac{1}{8}\right) -$$ -where the spin $K_i$'s satisfy the SU(1,1) algebra -\begin{eqnarray} -\label{commutatorsSU11} -[K_i^{0},K_i^{\pm}] &=& \pm K_i^{\pm} \nonumber \\ -{[}K_{i}^{-},K_{i}^{+}{]} &=& 2K_i^{0} -\end{eqnarray} -We are going to see the Schr\"odinger equation with imaginary time -\begin{equation} -\label{schroedinger} -\frac{d}{dt}|\psi(t) \rangle = -H |\psi(t)\rangle\;. -\end{equation} -as the evolution equation for the probability distribution of -a Markovian stochastic process. -\begin{itemize} -\item -\underline{The Hamiltonian possesses the SU(1,1) invariance}. If we define -\be -K^+ = \sum_{i} K_i^+ -\ee -\be -K^- = \sum_{i} K_i^- -\ee -\be -K^0 = \sum_{i} K_i^0 -\ee -we find that -\be -[H,K^+] = 0 -\ee -\be -[H,K^-] = 0 -\ee -\be -[H,K^0] = 0 -\ee -\item -\underline{Since $[H,K^+] = 0$} there exist a basis to study the stochastic process associated to -$H$ where \underline{$K^+$ is diagonal}. We might consider the following representation -\begin{eqnarray} -\label{Koper} -K^+_i &=& \frac{1}{2} x_{i}^2 \nonumber \\ -K^-_i &=& \frac{1}{2} \frac{\partial^2}{\partial x_{i}^2} \nonumber \\ -K^o_i &=& \frac{1}{4} \left\{\frac{\partial}{\partial x_{i}} x_{i} + - x_{i} \frac{\partial}{\partial x_{i}} \right \} -\end{eqnarray} -If we use this representation then -$$ -H = -L^* -$$ -and the probability density function for the $X(t)$ process is encoded in -the state $|\psi(t)\rangle$, namely -\begin{equation} -|\psi(t) \rangle = \int dx p(x,t) |x\rangle -\end{equation} -where we have introduced the notation $|x\rangle$ to denote a completely -localized state, that is a vector which together with its transposed -$\langle x|$ form a complete basis of a Hilbert space and its dual: -\begin{equation} -\langle x|x' \rangle = \delta(x-x') -\end{equation} -It immediately follows that -\begin{equation} -\langle x|\psi(t) \rangle = p(x,t) -\end{equation} -To compute expectation with respect to the $X(t)$ process -we introduce the flat state -\begin{equation} -\langle - | = \int dx \;\langle x| -\end{equation} -which is such that -\begin{equation} -\langle - | x\rangle = 1 -\end{equation} -Then for any observable $A = A(X(t))$ we have that its expectation value -at time $t$ can be written as -\begin{equation} -\langle A(t) \rangle_x = \int dy \,A(y)\, p(y,t;x,0) = \langle -|A| \psi(t) \rangle_x = \langle -|A e ^{-tH}| x\rangle -\end{equation} -\item -\underline{Since $[H,K^0] = 0$} there exist a basis to study the stochastic process associated to -$H$ where \underline{$K^0$ is diagonal}. We might consider the following representation -\begin{eqnarray} -\label{Koper2} -K^+_i|\xi\rangle &=& \left(\frac{1}{2} + \xi\right) |\xi+1\rangle\nonumber \\ -K^-_i|\xi\rangle &=& \xi |\xi-1\rangle\nonumber \\ -K^o_i|\xi\rangle &=& \left(\xi + \frac{1}{4}\right) |\xi\rangle -\end{eqnarray} -where $|\xi\rangle$ denotes a vector which together with its transposed -$\langle \xi|$ form a complete basis of a Hilbert space and its dual, that is -\begin{equation} -\langle \xi|\eta \rangle = \delta_{\xi,\eta} -\end{equation} -If we use this representation then -$$ -H = -{\cal L}^* -$$ -and the probability mass function for the $\Xi(t)$ process is encoded in -the state $|\phi(t)\rangle$, namely -\begin{equation} -|\phi(t) \rangle = \sum_{\xi} P(\xi,t) |\xi\rangle -\end{equation} -It immediately follows that -\begin{equation} -\langle \xi|\phi(t) \rangle = P(\xi,t) -\end{equation} -To compute expectation with respect to the $\Xi(t)$ process -we introduce the flat state -\begin{equation} -\langle -_{dual} | = \sum_{\xi} \;\langle \xi| -\end{equation} -which is such that -\begin{equation} -\langle -_{dual} | \xi\rangle = 1 -\end{equation} -Then for any observable $A=A(\Xi(t))$ we have that its expectation value -at time $t$ can be written as -\begin{equation} -\langle A(t) \rangle_\xi = \sum_{\eta}\,A(\eta)\, p(\eta,t;\xi,0) = \langle -_{dual}|A| \phi(t) \rangle_{\xi} = \langle -_{dual}|A e ^{-tH}| \xi\rangle -\end{equation} -\item -\underline{The claim is the following: Duality, in general, is going from the basis -where}\\ -\underline{one generator of the group is diagonal to a basis where another generator of}\\ -\underline{ the group is diagonal.} - -In our case we change from a basis where $K^+$ is diagonal to the base where $K^0$ is diagonal. - -\begin{eqnarray} -\langle - |\prod_i\frac{(2K_i^+)^{\xi_i}}{(2\xi_i-1)!!}|\psi(t)\rangle_x -& = & -\int dy \; \langle y |\prod_i\frac{(2K_i^+)^{\xi_i}}{(2\xi_i-1)!!} e^{tL^*}|x\rangle \nonumber \\ -& = & -\sum_{\eta} \int dy \; \langle y |\prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!}e^{tL^*}|x\rangle \langle \eta|\xi\rangle\nonumber \\ -& = & -\sum_{\eta} \int dy \; \langle y| \otimes \langle \eta| \prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!} e^{tL^*} | x\rangle \otimes|\xi\rangle\nonumber \\ -& = & -\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{tL} \prod_i\frac{(2K_i^+)^{\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta\rangle \nonumber \\ -& = & -\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{tL} \prod_i\frac{y^{2\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\ -& = & -\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{t{\cal L}} \prod_i\frac{y_i^{2\eta_i}}{(2\eta_i-1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\ -& = & -\sum_{\eta} \int dy \; \langle x| \otimes \langle \xi | e^{t{\cal L}} \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} | y\rangle \otimes|\eta \rangle \nonumber \\ -& = & -\sum_{\eta} \int dy \; \langle y| \otimes \langle \eta |\prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} e^{t{\cal L}^*} | x\rangle \otimes|\xi \rangle\nonumber \\ -& = & -\sum_{\eta} \int dy \; \langle \eta | \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} e^{t{\cal L}^*} |\xi \rangle \langle y | x\rangle \nonumber \\ -& = & -\sum_{\eta} \langle \eta | \prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!} |\phi(t)\rangle_{\xi} \nonumber \\ -& = & -\langle -_{dual} |\prod_i\frac{y_i^{K_i^0 -\frac{1}{2}\1}}{(K_i^0 -\frac{3}{2}\1)!!}|\phi(t)\rangle_{\xi} -\end{eqnarray} - - -\end{itemize} - -\section{General k} - -A convenient $(2j+1)$-dimensional representation of the SU(2) algebra is given by -\begin{eqnarray} -J^+_i |n_i\rangle &=& (2j-n_i) |n_i+1\rangle \nonumber \\ -J^-_i |n_i\rangle &=& n_i |n_i-1\rangle \nonumber \\ -J^0_i |n_i\rangle &=& (n_i-j) |n_i\rangle -\end{eqnarray} -where the quantum numbers $n_i\in\{0,1,\ldots,2j\}$. -{\bf Note that in this representation the adjoint of $J^+_i$ is not -$J^-_i$, UNLESS $j=1/2$}. - -A matrix representation is: -$$ -J^+ = \left( -\begin{array}{cccc} - 0 & & & \\ - 2j & \ddots & & \\ - & \ddots & \ddots & \\ - & & 1 & 0\\ -\end{array}\right) -\qquad -J^- = \left( -\begin{array}{cccc} - 0 & 1 & & \\ - & \ddots & \ddots & \\ - & & \ddots & 2j \\ - & & & 0 \\ -\end{array}\right) -\qquad -J^0 = \left( -\begin{array}{cccc} - -j & & & \\ - & \ddots & & \\ - & & \ddots & \\ - & & & j\\ -\end{array}\right) -$$ - -In the SU(1,1) case one can use the infinite dimensional representation -\begin{eqnarray} -\label{newrepresentationsu11} -K^+_i |n_i\rangle &=& (2k+n_i) |n_i+1\rangle \nonumber \\ -K^-_i |n_i\rangle &=& n_i |n_i-1\rangle \nonumber \\ -K^0_i |n_i\rangle &=& (n_i+k) |n_i\rangle -\end{eqnarray} -where the quantum numbers $n_i\in\{0,1,2,\ldots\}$. -A matrix representation is: -$$ -K^+ = \left( -\begin{array}{cccc} - 0 & & & \\ - 2k & \ddots & & \\ - & 2k+1 & \ddots & \\ - & & \ddots & \ddots\\ -\end{array}\right) -\qquad -K^- = \left( -\begin{array}{cccc} - 0 & 1 & & \\ - & \ddots & 2 & \\ - & & \ddots & \ddots \\ - & & & \ddots \\ -\end{array}\right) -\qquad -K^0 = \left( -\begin{array}{cccc} - k & & & \\ - & k+1 & & \\ - & & k+2 & \\ - & & & \ddots\\ -\end{array}\right) -$$ -Let's check that in this representation the operator is stochastic. -I will do it for the bulk: -\begin{eqnarray} -L_{i,i+1}|n_i,n_{i+1}\rangle -&=& -(2k+n_i) n_{i+1}|n_i +1 ,n_{i+1}-1\rangle \nonumber\\ -&+& -n_i(2k+n_{i+1})|n_i -1 ,n_{i+1}+1\rangle \nonumber\\ -&+& -(-2(n_i+k)(n_{i+1}+k)+2k^2)|n_i,n_{i+1}\rangle -\end{eqnarray} -The sum of the rates is -$$ -(2k+n_i) n_{i+1}+ -n_i(2k+n_{i+1}) --2(n_i+k)(n_{i+1}+k)+2k^2 =0 -$$ - - - - - - - - - -% \end{document} -- cgit v1.2.3-54-g00ecf From 274943e723eb96f8c53a4abf5e090befa66f9cca Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 14:01:29 +0100 Subject: Switched to biblatex. --- cover.tex | 46 ++++++++++++++++------------------------------ 1 file changed, 16 insertions(+), 30 deletions(-) diff --git a/cover.tex b/cover.tex index 2d2fc10..ec9ff7a 100644 --- a/cover.tex +++ b/cover.tex @@ -1,25 +1,5 @@ \documentclass[a4paper]{letter} -\makeatletter -\newenvironment{thebibliography}[1] - {\list{\@biblabel{\@arabic\c@enumiv}}% - {\settowidth\labelwidth{\@biblabel{#1}}% - \leftmargin\labelwidth - \advance\leftmargin\labelsep - \usecounter{enumiv}% - \let\p@enumiv\@empty - \renewcommand\theenumiv{\@arabic\c@enumiv}}% - \sloppy - \clubpenalty4000 - \@clubpenalty \clubpenalty - \widowpenalty4000% - \sfcode`\.\@m} - {\def\@noitemerr - {\@latex@warning{Empty `thebibliography' environment}}% - \endlist} -\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em} -\makeatother - \usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? \usepackage[T1]{fontenc} % vector fonts plz \usepackage{newtxtext,newtxmath} % Times for PR @@ -31,8 +11,12 @@ linkcolor=purple ]{hyperref} % ref and cite links with pretty colors \usepackage{xcolor} +\usepackage[style=phys]{biblatex} + +\addbibresource{bezout.bib} \signature{ + \vspace{-3.5em} Jaron Kent-Dobias \& Jorge Kurchan } @@ -53,16 +37,16 @@ \opening{} -The subject of `Complex Landscapes,' which started in the spin-glass +The subject of `complex landscapes,' which started in the spin-glass literature, is concerned with functions (landscapes) of many variables, having -a multiplicity of minimums, which are the objects of interest. Apart from its -obvious interest for glassy systems, it has found a myriad applications in -many domains: Computer Science, Ecology, Economics, Biology +a multiplicity of minima, which are the objects of interest. Apart from its +obvious interest for glassy systems, it has found a myriad applications in many +domains: computer science, ecology, economics, biology \cite{Mezard_2009_Information}. In the last few years, a renewed interest has developed for landscapes for which the variables are complex. There are a few reasons for this: {\em i)} in -Computational Physics, there is the main obstacle of the `sign problem', and a +computational physics, there is the main obstacle of the `sign problem', and a strategy has emerged to attack it deforming the sampling space into complex variables. This is a most natural and promising path, and any progress made will have game-changing impact in solid state physics and lattice-QCD @@ -73,17 +57,19 @@ concerning the very definition of quantum mechanics, which requires also that one move into the complex plane. In all these cases, just like in the real case, one needs to know the structure -of the `landscape', where are the saddle points and how they are connected, -typical questions of `complexity'. However, to the best of our knowledge, +of the `landscape.' where are the saddle points and how they are connected, +typical questions of `complexity.' However, to the best of our knowledge, there are no studies extending the methods of the theory of complexity to complex variables. We believe our paper will open a field that may find numerous applications and will widen our theoretical view of complexity in general. \closing{Sincerely,} -\end{letter} -\bibliographystyle{unsrt} -\bibliography{bezout} +\vspace{1em} + +\printbibliography[heading=none] + +\end{letter} \end{document} -- cgit v1.2.3-54-g00ecf From 9f0171b52574d8e63fc9178d36b41e78346e5805 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 14:09:54 +0100 Subject: Fancy footnote citations. --- cover.tex | 14 ++++++-------- 1 file changed, 6 insertions(+), 8 deletions(-) diff --git a/cover.tex b/cover.tex index ec9ff7a..a20ec7f 100644 --- a/cover.tex +++ b/cover.tex @@ -41,18 +41,18 @@ The subject of `complex landscapes,' which started in the spin-glass literature, is concerned with functions (landscapes) of many variables, having a multiplicity of minima, which are the objects of interest. Apart from its obvious interest for glassy systems, it has found a myriad applications in many -domains: computer science, ecology, economics, biology -\cite{Mezard_2009_Information}. +domains: computer science, ecology, economics, biology. +\footfullcite{Mezard_2009_Information} In the last few years, a renewed interest has developed for landscapes for which the variables are complex. There are a few reasons for this: {\em i)} in computational physics, there is the main obstacle of the `sign problem', and a strategy has emerged to attack it deforming the sampling space into complex variables. This is a most natural and promising path, and any progress made -will have game-changing impact in solid state physics and lattice-QCD -\cite{Cristoforetti_2012_New,Scorzato_2016_The}. {\em ii)} At a more basic -level, following the seminal work of E. Witten -\cite{Witten_2010_A,Witten_2011_Analytic}, there has been a flurry of activity +will have game-changing impact in solid state physics and lattice-QCD. +\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} {\em ii)} At a more basic +level, following the seminal work of E.~Witten, +\footfullcite{Witten_2010_A,Witten_2011_Analytic} there has been a flurry of activity concerning the very definition of quantum mechanics, which requires also that one move into the complex plane. @@ -68,8 +68,6 @@ general. \vspace{1em} -\printbibliography[heading=none] - \end{letter} \end{document} -- cgit v1.2.3-54-g00ecf From 5352b6f5885f1d46dfd16d8e998e9ed4a42f76c1 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 14:13:03 +0100 Subject: Added salutation. --- cover.tex | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/cover.tex b/cover.tex index a20ec7f..3c90986 100644 --- a/cover.tex +++ b/cover.tex @@ -16,14 +16,15 @@ \addbibresource{bezout.bib} \signature{ - \vspace{-3.5em} + \vspace{-6\medskipamount} + \smallskip Jaron Kent-Dobias \& Jorge Kurchan } \address{ Laboratoire de Physique \\ Ecole Normale Sup\'erieure \\ - 24, rue Lhomond \\ + 24 rue Lhomond \\ 75005 Paris } @@ -35,7 +36,7 @@ Ridge, NY 11961 } -\opening{} +\opening{To the editors of Physical Review,} The subject of `complex landscapes,' which started in the spin-glass literature, is concerned with functions (landscapes) of many variables, having -- cgit v1.2.3-54-g00ecf From 47e2da9200806b26d3da4ef94f7ac4cd1355705c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 14:13:34 +0100 Subject: New .gitignore entries for biblatex files. --- .gitignore | 2 ++ 1 file changed, 2 insertions(+) diff --git a/.gitignore b/.gitignore index bc9e69d..9edb78b 100644 --- a/.gitignore +++ b/.gitignore @@ -10,3 +10,5 @@ *.dvi *.synctex.gz *.synctex(busy) +*.bcf +*.run.xml -- cgit v1.2.3-54-g00ecf From 8373f749bbc33d63bfa2993cdba0c3dceb74fe50 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 14:15:51 +0100 Subject: Small bib fixes. --- bezout.bib | 4 ++-- cover.tex | 2 +- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/bezout.bib b/bezout.bib index 9e600f0..ba871fb 100644 --- a/bezout.bib +++ b/bezout.bib @@ -314,7 +314,7 @@ volume = {251}, url = {https://doi.org/10.22323%2F1.251.0016}, doi = {10.22323/1.251.0016}, - booktitle = {Proceedings of The 33rd International Symposium on Lattice Field Theory (LATTICE 2015)}, + booktitle = {Proceedings of The 33rd International Symposium on Lattice Field Theory ({LATTICE} 2015)}, series = {Proceedings of Science} } @@ -369,7 +369,7 @@ pages = {347--446}, url = {https://doi.org/10.1090%2Famsip%2F050%2F19}, doi = {10.1090/amsip/050/19}, - booktitle = {Chern-Simons Gauge Theory: 20 Years After}, + booktitle = {{Chern}-{Simons} Gauge Theory: 20 Years After}, editor = {Andersen, Jørgen E. and Boden, Hans U. and Hahn, Atle and Himpel, Benjamin}, series = {AMS/IP Studies in Advanced Mathematics} } diff --git a/cover.tex b/cover.tex index 3c90986..c721c52 100644 --- a/cover.tex +++ b/cover.tex @@ -53,7 +53,7 @@ variables. This is a most natural and promising path, and any progress made will have game-changing impact in solid state physics and lattice-QCD. \footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} {\em ii)} At a more basic level, following the seminal work of E.~Witten, -\footfullcite{Witten_2010_A,Witten_2011_Analytic} there has been a flurry of activity +\footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a flurry of activity concerning the very definition of quantum mechanics, which requires also that one move into the complex plane. -- cgit v1.2.3-54-g00ecf From cf3f83827d960a70383232b65bef72398646fc10 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 14:20:57 +0100 Subject: Made citations links invisible. --- cover.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/cover.tex b/cover.tex index c721c52..b414930 100644 --- a/cover.tex +++ b/cover.tex @@ -6,9 +6,9 @@ \usepackage[ colorlinks=true, urlcolor=purple, - citecolor=purple, - filecolor=purple, - linkcolor=purple + linkcolor=black, + citecolor=black, + filecolor=black ]{hyperref} % ref and cite links with pretty colors \usepackage{xcolor} \usepackage[style=phys]{biblatex} -- cgit v1.2.3-54-g00ecf From 50767662937073c6804888dd25ab49420c7a939e Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 14:30:29 +0100 Subject: Wording edits. --- cover.tex | 37 ++++++++++++++++++------------------- 1 file changed, 18 insertions(+), 19 deletions(-) diff --git a/cover.tex b/cover.tex index b414930..6e21d10 100644 --- a/cover.tex +++ b/cover.tex @@ -39,27 +39,26 @@ \opening{To the editors of Physical Review,} The subject of `complex landscapes,' which started in the spin-glass -literature, is concerned with functions (landscapes) of many variables, having -a multiplicity of minima, which are the objects of interest. Apart from its -obvious interest for glassy systems, it has found a myriad applications in many -domains: computer science, ecology, economics, biology. -\footfullcite{Mezard_2009_Information} +literature, is concerned with functions (landscapes) of many variables having +a multiplicity of minima. Apart from its obvious relevance to glassy systems, +it has found applications in many domains: computer science, ecology, +economics, and biology, to name a few. \footfullcite{Mezard_2009_Information} -In the last few years, a renewed interest has developed for landscapes for -which the variables are complex. There are a few reasons for this: {\em i)} in -computational physics, there is the main obstacle of the `sign problem', and a -strategy has emerged to attack it deforming the sampling space into complex -variables. This is a most natural and promising path, and any progress made -will have game-changing impact in solid state physics and lattice-QCD. -\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} {\em ii)} At a more basic -level, following the seminal work of E.~Witten, -\footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a flurry of activity -concerning the very definition of quantum mechanics, which requires also that -one move into the complex plane. +In the last few years, a renewed interest has developed in landscapes for +which the variables are complex. There are a couple reasons for this: in +computational physics the `sign problem' is a major obstacle, and a strategy +has emerged to attack it by deforming the sampling space into complex variables. +This is a most natural and promising path, and any progress made will have +game-changing impact in solid state physics and lattice QCD. +\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} At a more basic +level, following the seminal work of E.~Witten, \footfullcite{Witten_2010_A, +Witten_2011_Analytic} there has been a flurry of activity concerning the very +definition of quantum mechanics, which also requires that one move into the +complex plane. -In all these cases, just like in the real case, one needs to know the structure -of the `landscape.' where are the saddle points and how they are connected, -typical questions of `complexity.' However, to the best of our knowledge, +In these cases, just as in the real case, one needs to understand the structure +of the `landscape,' like the location of saddle points, how they are connected, +and typical questions of `complexity.' However, to the best of our knowledge, there are no studies extending the methods of the theory of complexity to complex variables. We believe our paper will open a field that may find numerous applications and will widen our theoretical view of complexity in -- cgit v1.2.3-54-g00ecf From 569cd75da9eb1de4adbd8d2294cb9b3686923a82 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 14:33:10 +0100 Subject: More wording edits. --- cover.tex | 21 ++++++++++----------- 1 file changed, 10 insertions(+), 11 deletions(-) diff --git a/cover.tex b/cover.tex index 6e21d10..6f74b46 100644 --- a/cover.tex +++ b/cover.tex @@ -44,17 +44,16 @@ a multiplicity of minima. Apart from its obvious relevance to glassy systems, it has found applications in many domains: computer science, ecology, economics, and biology, to name a few. \footfullcite{Mezard_2009_Information} -In the last few years, a renewed interest has developed in landscapes for -which the variables are complex. There are a couple reasons for this: in -computational physics the `sign problem' is a major obstacle, and a strategy -has emerged to attack it by deforming the sampling space into complex variables. -This is a most natural and promising path, and any progress made will have -game-changing impact in solid state physics and lattice QCD. -\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} At a more basic -level, following the seminal work of E.~Witten, \footfullcite{Witten_2010_A, -Witten_2011_Analytic} there has been a flurry of activity concerning the very -definition of quantum mechanics, which also requires that one move into the -complex plane. +Recently, interest has developed in landscapes for which the variables are +complex. There are several reasons for this: in computational physics the +`sign problem' is a major obstacle, and a strategy has emerged to attack it by +deforming the sampling space into complex variables. This is a most natural +and promising path, and any progress made will have game-changing impact in +solid state physics and lattice QCD. \footfullcite{Cristoforetti_2012_New, +Scorzato_2016_The} At a more basic level, following the seminal work of +E.~Witten, \footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a +flurry of activity concerning the very definition of quantum mechanics, which +also requires that one move into the complex plane. In these cases, just as in the real case, one needs to understand the structure of the `landscape,' like the location of saddle points, how they are connected, -- cgit v1.2.3-54-g00ecf From c3aef8d99961ccc06df11a4369bf112f483cc8cd Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 15:13:52 +0100 Subject: Added name to return address. --- cover.tex | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/cover.tex b/cover.tex index 6f74b46..5abbdfe 100644 --- a/cover.tex +++ b/cover.tex @@ -22,9 +22,10 @@ } \address{ - Laboratoire de Physique \\ - Ecole Normale Sup\'erieure \\ - 24 rue Lhomond \\ + Jaron Kent-Dobias\\ + Laboratoire de Physique\\ + Ecole Normale Sup\'erieure\\ + 24 rue Lhomond\\ 75005 Paris } -- cgit v1.2.3-54-g00ecf From d13787cf9f35ab04eaf961f55cfbb8f1c75e423f Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Dec 2020 15:43:29 +0100 Subject: Added sentence noted that the manuscript was modified. --- cover.tex | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/cover.tex b/cover.tex index 5abbdfe..4ebc19b 100644 --- a/cover.tex +++ b/cover.tex @@ -22,7 +22,6 @@ } \address{ - Jaron Kent-Dobias\\ Laboratoire de Physique\\ Ecole Normale Sup\'erieure\\ 24 rue Lhomond\\ @@ -62,7 +61,7 @@ and typical questions of `complexity.' However, to the best of our knowledge, there are no studies extending the methods of the theory of complexity to complex variables. We believe our paper will open a field that may find numerous applications and will widen our theoretical view of complexity in -general. +general. Our manuscript has been amended to emphasize these important connections with other areas of physics. \closing{Sincerely,} -- cgit v1.2.3-54-g00ecf