From a19e159544b27f087e319f228f45c68546d6bde6 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Thu, 10 Dec 2020 12:49:12 +0000
Subject: Update on Overleaf.
---
bezout.tex | 27 +++++++++++++++------------
1 file changed, 15 insertions(+), 12 deletions(-)
diff --git a/bezout.tex b/bezout.tex
index 6fc863c..debb93c 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -53,10 +53,9 @@ defined by the energy:
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
+study here. This problem has been studied 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
+It 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
@@ -66,13 +65,13 @@ 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
+In this paper we shall extend the study to the case where the variables are complex
+$z\in\mathbb C^N$ 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
+are indeed situations in which complex variables 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
@@ -85,9 +84,9 @@ 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
+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
@@ -97,12 +96,11 @@ 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 $z^2=N$,
+ For our model the constraint we choose $z^2=N$,
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:
+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).
@@ -110,6 +108,11 @@ introducing the $\epsilon\in\mathbb C$, this gives
It is easy to see that {\em for a pure $p$-spin}, at any critical point
$\epsilon=H/N$, the average energy.
+Critical points are given by the set of equations:
+
+\begin{equation}
+\frac{c_p}{(p-1)!}\sum_{ i, i_2\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
+
Since $H$ is holomorphic, a point is a critical point of its real part if and
only if it is also a critical point of its imaginary part. The number of
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