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.tex | 50 ++++++++++++++++++++++++++++++++------------------
 1 file changed, 32 insertions(+), 18 deletions(-)

(limited to 'bezout.tex')

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.
 
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