Merge branch 'master' into apsaps.v2

5 files changed, 221 insertions, 40 deletions

diff --git a/.gitignore b/.gitignore
index bc9e69d..9edb78b 100644
--- a/.gitignore
+++ b/.gitignore
@@ -10,3 +10,5 @@
 *.dvi
 *.synctex.gz
 *.synctex(busy)
+*.bcf
+*.run.xml
diff --git a/bezout.bib b/bezout.bib
index 022f5ea..ba871fb 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -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},
@@ -92,7 +105,7 @@
 
 @article{Bray_2007_Statistics,
  author = {Bray, Alan J. and Dean, David S.},
- title = {Statistics of Critical Points of Gaussian Fields on Large-Dimensional Spaces},
+ title = {Statistics of Critical Points of {Gaussian} Fields on Large-Dimensional Spaces},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2007},
@@ -148,7 +161,7 @@
 
 @article{Crisanti_1995_Thouless-Anderson-Palmer,
  author = {Crisanti, A. and Sommers, H.-J.},
- title = {Thouless-Anderson-Palmer Approach to the Spherical p-Spin Spin Glass Model},
+ title = {{Thouless}-{Anderson}-{Palmer} Approach to the Spherical p-Spin Spin Glass Model},
  journal = {Journal de Physique I},
  publisher = {EDP Sciences},
  year = {1995},
@@ -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},
@@ -176,7 +203,7 @@
 
 @article{Dyson_1962_A,
  author = {Dyson, Freeman J.},
- title = {A Brownian-Motion Model for the Eigenvalues of a Random Matrix},
+ title = {A {Brownian}-Motion Model for the Eigenvalues of a Random Matrix},
  journal = {Journal of Mathematical Physics},
  publisher = {AIP Publishing},
  year = {1962},
@@ -240,6 +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},
@@ -268,6 +305,33 @@
  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)},
@@ -282,4 +346,32 @@
  doi = {10.1007/bf01456804}
 }
 
+@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}
+}
+
+@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 d7f33df..8e354da 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{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}
@@ -72,35 +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 they 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
-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:
-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}
@@ -108,12 +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
@@ -444,18 +445,23 @@ 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
-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
-threshold level, where the system develops a mid-spectrum gap, will play a
-crucial role as it does in the real case.
+ 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, 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
+ 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.
   JK-D and JK are supported by the Simons Foundation Grant No.~454943.
 \end{acknowledgments}
 
-\bibliographystyle{apsrev4-2}
 \bibliography{bezout}
 
 \end{document}
diff --git a/cover.tex b/cover.tex
new file mode 100644
index 0000000..4ebc19b
--- /dev/null
+++ b/cover.tex
@@ -0,0 +1,72 @@
+\documentclass[a4paper]{letter}
+
+\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,
+  linkcolor=black,
+  citecolor=black,
+  filecolor=black
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{xcolor}
+\usepackage[style=phys]{biblatex}
+
+\addbibresource{bezout.bib}
+
+\signature{
+  \vspace{-6\medskipamount}
+  \smallskip
+  Jaron Kent-Dobias \& Jorge Kurchan
+}
+
+\address{
+  Laboratoire de Physique\\
+  Ecole Normale Sup\'erieure\\
+  24 rue Lhomond\\ 
+  75005 Paris
+}
+
+\begin{document}
+\begin{letter}{
+  Editorial Office\\
+  Physical Review Letters\\
+  1 Research Road\\
+  Ridge, NY 11961
+}
+
+\opening{To the editors of Physical Review,}
+
+The subject of `complex landscapes,' which started in the spin-glass
+literature, is concerned with functions (landscapes) of many variables having
+a multiplicity of minima. Apart from its obvious relevance to glassy systems,
+it has found applications in many domains: computer science, ecology,
+economics, and biology, to name a few.  \footfullcite{Mezard_2009_Information}
+
+Recently, interest has developed in landscapes for which the variables are
+complex. There are several reasons for this: in computational physics the
+`sign problem' is a major obstacle, and a strategy has emerged to attack it by
+deforming the sampling space into complex variables.  This is a most natural
+and promising path, and any progress made will have game-changing impact in
+solid state physics and lattice QCD.  \footfullcite{Cristoforetti_2012_New,
+Scorzato_2016_The} At a more basic level, following the seminal work of
+E.~Witten, \footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a
+flurry of activity concerning the very definition of quantum mechanics, which
+also requires that one move into the complex plane. 
+
+In these cases, just as in the real case, one needs to understand the structure
+of the `landscape,' like the location of saddle points, how they are connected,
+and typical questions of `complexity.'  However, to the best of our knowledge,
+there are no studies extending the methods of the theory of complexity to
+complex variables.  We believe our paper will open a field that may find
+numerous applications and will widen our theoretical view of complexity in
+general. Our manuscript has been amended to emphasize these important connections with other areas of physics.
+
+\closing{Sincerely,}
+
+\vspace{1em}
+
+\end{letter}
+
+\end{document}
diff --git a/why_prl.txt b/why_prl.txt
new file mode 100644
index 0000000..8c0eef1
--- /dev/null
+++ b/why_prl.txt
@@ -0,0 +1,9 @@
+We extend the paradigmatic model of "complex" landscapes to complex variables.
+We believe it is the first such study, a subject of interest with applications
+from deep networks to optimization. In particular, we introduce an apparently
+new matrix model that generalizes the well-known semicircle law for
+fluctuations around real saddles in disordered systems. Our work is in line
+with Bogomolny, Bohigas & Leboeuf (PRL 1992) concerning roots of one random
+polynomial of high degree, while ours many of low degree. Many applications of
+this problem will surely appear, as always occurs when extending a real problem
+into the complex plane.