From ad1ec8fc25dc3581e6c2cfa7cc119d47e92a2185 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 9 Dec 2020 17:46:11 +0100
Subject: Added some spaces and reflowed the text to enfornce shorter lines.

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 \date\today
 
 \begin{abstract}
-  We study the saddle-points of the $p$-spin model -- the best understood example of `complex (rugged) landscape' -- in the space in which all its $N$ variables are allowed to be complex. The problem becomes
-  a system of $N$ random equations of degree $p-1$. 
-  We solve for quantities averaged over randomness in the $N \rightarrow \infty$ limit. 
-  We show that the number of solutions saturates the Bézout bound $\ln {\cal{N}}\sim N \ln (p-1) $\cite{Bezout_1779_Theorie}.
-The Hessian of each saddle is given by a random matrix of the form $M=a A^2+b B^2 +ic [A,B]_-+ d [A,B]_+$,
-where $A$ and $B$ are GOE matrices and $a-d$ real. Its spectrum  has a transition 
-from one-cut to two-cut that generalizes the notion  of `threshold level' that is well-known in the real problem.
-In the case that the disorder is itself real, only the square-root of the total  number solutions are real.
-In terms of real and imaginary parts of the energy, the solutions are divided in  sectors where the saddles have
-different topological properties.
+  We study the saddle-points of the $p$-spin model -- the best understood
+  example of `complex (rugged) landscape' -- in the space in which all its $N$
+  variables are allowed to be complex. The problem becomes a system of $N$
+  random equations of degree $p-1$.  We solve for quantities averaged over
+  randomness in the $N \rightarrow \infty$ limit.  We show that the number of
+  solutions saturates the Bézout bound $\ln {\cal{N}}\sim N \ln (p-1)$
+  \cite{Bezout_1779_Theorie}.  The Hessian of each saddle is given by a random
+  matrix of the form $M=a A^2+b B^2 +ic [A,B]_-+ d [A,B]_+$, where $A$ and $B$
+  are GOE matrices and $a-d$ real. Its spectrum  has a transition from one-cut
+  to two-cut that generalizes the notion  of `threshold level' that is
+  well-known in the real problem.  In the case that the disorder is itself
+  real, only the square-root of the total  number solutions are real.  In terms
+  of real and imaginary parts of the energy, the solutions are divided in
+  sectors where the saddles have different topological properties.
 \end{abstract}
 
 \maketitle
 
-Spin-glasses have long been considered the paradigm of `complex landscapes' of many variables, a subject that
-includes Neural Networks and optimization problems, most notably  Constraint Satisfaction ones.
-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})
+Spin-glasses have long been considered the paradigm of `complex landscapes' of
+many variables, a subject that includes Neural Networks and optimization
+problems, most notably  Constraint Satisfaction ones.  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}
   H_0 = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
-where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are real and constrained
-to a sphere $\sum_i z_i^2=N$. If there is a single  term of a given $p$, this is known as the `pure $p$-spin' model, the case we shall study here.
-Also in the Algebra \cite{Cartwright_2013_The} and Probability  literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}.
+where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are
+real and constrained to a sphere $\sum_i z_i^2=N$. If there is a single  term
+of a given $p$, this is known as the `pure $p$-spin' model, the case we shall
+study here.  Also in the Algebra \cite{Cartwright_2013_The} and Probability
+literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}.
 
 This problem has been attacked from several angles: the replica trick to
-compute the Boltzmann--Gibbs distribution\cite{Crisanti_1992_The}, a Kac--Rice \cite{Kac_1943_On,
-Rice_1939_The, Fyodorov_2004_Complexity} procedure (similar to the Fadeev--Popov
-integral)  to compute the number of saddle-points of the energy function
-\cite{Crisanti_1995_Thouless-Anderson-Palmer}, and
-the gradient-descent -- or more generally Langevin -- dynamics staring from a
-high-energy configuration \cite{Cugliandolo_1993_Analytical}.  Thanks to the relative simplicity of the energy,
-all these approaches are possible analytically in the large $N$ limit.
-
-In this paper we shall extend the study to the case  where $z\in\mathbb C^N$ are  and $J$ is a symmetric tensor
-whose elements are complex normal with $\overline{|J|^2}=p!/2N^{p-1}$ and
-$\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$.
-
-The motivations for this paper are of two types. On the practical side, there are situations in which  complex variables
-have in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Antenucci_2015_Complex}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos_2016_Disordered} modeling Black Hole horizons in the zero-temperature limit.  
-
-There is however a more fundamental reason for this study: we know from experience that extending a problem to the complex plane often uncovers an underlying simplicity that is hidden in the purely real case. 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 roots. Because we are working in
-complex variables, and the roots are simple all the way (we shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. 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.
-
-This study also provides a complement to the work on the distribution of zeroes of random polynomials \cite{Bogomolny_1992_Distribution}.
-
-
-Let us go back to our model.
-For the  constraint we  choose here $z^2=N$, rather than $|z|^2=N$, in order to preserve the holomorphic nature
-of the functions. In addition, the  
-nonholomorphic spherical constraint has a disturbing lack of critical points
-nearly everywhere, since $0=\partial^* H=-p\epsilon z$ is only
-satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint.
-It is enforced using the method of Lagrange
-multipliers: introducing the $\epsilon\in\mathbb C$, this gives
+compute the Boltzmann--Gibbs distribution \cite{Crisanti_1992_The}, a Kac--Rice
+\cite{Kac_1943_On, Rice_1939_The, Fyodorov_2004_Complexity} procedure (similar
+to the Fadeev--Popov integral)  to compute the number of saddle-points of the
+energy function \cite{Crisanti_1995_Thouless-Anderson-Palmer}, and the
+gradient-descent -- or more generally Langevin -- dynamics staring from a
+high-energy configuration \cite{Cugliandolo_1993_Analytical}.  Thanks to the
+relative simplicity of the energy, all these approaches are possible
+analytically in the large $N$ limit.
+
+In this paper we shall extend the study to the case  where $z\in\mathbb C^N$
+are  and $J$ is a symmetric tensor whose elements are complex normal with
+$\overline{|J|^2}=p!/2N^{p-1}$ and $\overline{J^2}=\kappa\overline{|J|^2}$ for
+complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$.
+
+The motivations for this paper are of two types. On the practical side, there
+are situations in which  complex variables have in a disorder problem appear
+naturally: such is the case in which they are {\em phases}, as in random laser
+problems \cite{Antenucci_2015_Complex}. Another problem where a Hamiltonian
+very close to ours has been proposed is the Quiver Hamiltonians
+\cite{Anninos_2016_Disordered} modeling Black Hole horizons in the
+zero-temperature limit.  
+
+There is however a more fundamental reason for this study: we know from
+experience that extending a problem to the complex plane often uncovers an
+underlying simplicity that is hidden in the purely real case. 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 roots. Because we
+are working in complex variables, and the roots are simple all the way (we
+shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. 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.
+
+This study also provides a complement to the work on the distribution of zeroes
+of random polynomials \cite{Bogomolny_1992_Distribution}.
+
+
+Let us go back to our model.  For the  constraint we  choose here $z^2=N$,
+rather than $|z|^2=N$, in order to preserve the holomorphic nature of the
+functions. In addition, the  nonholomorphic spherical constraint has a
+disturbing lack of critical points nearly everywhere, since $0=\partial^*
+H=-p\epsilon z$ is only satisfied for $\epsilon=0$, as $z=0$ is forbidden by
+the constraint.  It is enforced using the method of Lagrange multipliers:
+introducing the $\epsilon\in\mathbb C$, this gives
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
-It is easy to see that {\em for a pure $p$-spin}, at  any critical point $\epsilon=H/N$, the average energy.
+It is easy to see that {\em for a pure $p$-spin}, at  any critical point
+$\epsilon=H/N$, the average energy.
 
 
 Since $H$ is holomorphic, a point is a critical point of its real part if and
-- 
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