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.
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