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| -rw-r--r-- | bezout.bib | 129 | ||||
| -rw-r--r-- | bezout.tex | 4 |
2 files changed, 81 insertions, 52 deletions
@@ -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} } @@ -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. |