From 7ffd9c8ac7df8bf7d7ff3f469277aa7439b8db06 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Dec 2020 20:59:31 +0100
Subject: Found stray references.

---
 bezout.tex | 14 +++++++++-----
 1 file changed, 9 insertions(+), 5 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index caa634a..83e748e 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -46,16 +46,20 @@ The most tractable family of these  are the mean-field spherical p-spin models d
 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.
 
-This problem has been attacked from several angles: the replica trick to compute the Boltzmann-Gibbs distribution,
-a Kac-Rice \cite{Kac,Fyodorov} procedure (similar to the Fadeev-Popov integral)  to compute the number of saddle-points of the energy function, and the gradient-descent -- or more generally Langevin -- dynamics staring from a high-energy configuration.
-Thanks to the relative simplicity of the energy, all these approaches are possible analytically in the large $N$ limit.
+This problem has been attacked from several angles: the replica trick to
+compute the Boltzmann--Gibbs distribution, 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, and
+the gradient-descent -- or more generally Langevin -- dynamics staring from a
+high-energy configuration.  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{Leuzzi-Crisanti, etc}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos-Anous-Denef} modeling Black Hole horizons in the zero-temperature limit.  
+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, etc}. 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 
@@ -98,7 +102,7 @@ $N \Sigma=
 \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In most, but not all,  of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim 
 \ln \overline{ \mathcal N_J} = \ln \int dJ \; N_J$ is exact.
 
-A useful property of the Gaussian distributions is that gradient and Hessian may be seen to be independent \cite{BrayDean,Fyodorov},
+A useful property of the Gaussian distributions is that gradient and Hessian may be seen to be independent \cite{Bray_2007_Statistics, Fyodorov_2004_Complexity},
 so that we may treat the delta-functions and the Hessians as independent.
 
 }
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