From dd2e5767e8b7e63c5210fb4dad3ad5b5cf6fff81 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 24 Dec 2020 09:50:54 +0100
Subject: Removed incorrectly placed bibliographystyle directive and fixed
 capitalization of titles.

---
 bezout.tex | 1 -
 1 file changed, 1 deletion(-)

(limited to 'bezout.tex')

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 v1.3


From d9701957da92a97eda685ac44864b12d665a285d Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
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(-)

(limited to 'bezout.tex')

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 v1.3


From 7bc5969c319a760c3259455e5bd24d1694ba1def Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
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(-)

(limited to 'bezout.tex')

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 v1.3


From 9cec3f1cff603521aa4e948576b56f06d5902f39 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Dec 2020 19:46:57 +0100
Subject: English and paragraph quadriture.

---
 bezout.tex | 89 +++++++++++++++++++++++++++++---------------------------------
 1 file changed, 41 insertions(+), 48 deletions(-)

(limited to 'bezout.tex')

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 v1.3


From 3aa483b31c12d6a3afb1215ce6188aca8c314e72 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
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

(limited to 'bezout.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 @@
+
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+
+\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}
+<O> = \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}
+<O> = \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}
+<O> &=& \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}
+<O> = \langle - | O e^{-Ht} | init \rangle
+\end{equation}
+where $\langle - |$ is a constant.
+As before:
+\begin{eqnarray}
+<O> &=& \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}
+<O> &=& \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}
+<O> &=&  \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 v1.3


From 57c597f34858eb256a1c681ab931cf0ec665cec3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
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(-)

(limited to 'bezout.tex')

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 @@
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+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\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
 }
-% 
 
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+\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}
-<O> = \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}
-<O> = \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}
-<O> &=& \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}
-<O> = \langle - | O e^{-Ht} | init \rangle
-\end{equation}
-where $\langle - |$ is a constant.
-As before:
-\begin{eqnarray}
-<O> &=& \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}
-<O> &=& \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}
-<O> &=&  \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}
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