From d0945d932b815a7eeab00723d4b76d32d037216f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Dec 2020 17:13:40 +0100
Subject: More small changes.

---
 bezout.tex | 21 ++++++++++-----------
 1 file changed, 10 insertions(+), 11 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 57e48ae..8c24d59 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -48,12 +48,10 @@ 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},
+  H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
 where $J$ is a symmetric tensor whose elements are real Gaussian variables and
-$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. 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 studied in the algebra
+$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. This problem has been studied in the algebra
 \cite{Cartwright_2013_The} and probability literature
 \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}.  It has been attacked
 from several angles: the replica trick to compute the Boltzmann--Gibbs
@@ -99,7 +97,7 @@ 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}
-For a \emph{pure} $p$-spin, $\epsilon=H/N$ -- the average energy -- at any
+It is easily shown that $\epsilon=H/N$ -- the average energy -- at any
 critical point. We choose to constrain our model by $z^2=N$ rather than
 $|z|^2=N$ in order to preserve the holomorphic nature of $H$. In addition, the
 nonholomorphic spherical constraint has a disturbing lack of critical points
@@ -241,8 +239,9 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of the unconstr
   \partial_i\partial_jH_0
   =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}},
 \end{equation}
-which makes its ensemble that of Gaussian complex symmetric matrices. Given its
-variances $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
+which makes its ensemble that of Gaussian complex symmetric matrices when the
+direction along the constraint is neglected. Given its variances
+$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
 $\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, its distribution of
 eigenvalues $\rho_0(\lambda)$ is constant inside the ellipse
 \begin{equation} \label{eq:ellipse}
@@ -389,7 +388,7 @@ contained within is
   = \frac12N\log(p-1),
 \end{equation}
 half of \eqref{eq:bezout} and corresponding precisely to the number of critical
-points of the real pure spherical $p$-spin model. (note the role of conjugation
+points of the real $p$-spin model. (note the role of conjugation
 symmetry, already underlined in \cite{Bogomolny_1992_Distribution}). The full
 $\epsilon$-dependence of the real $p$-spin is recovered by this limit as
 $\epsilon$ is varied.
@@ -398,7 +397,7 @@ $\epsilon$ is varied.
   \centering
   \includegraphics{fig/complexity.pdf}
   \caption{
-    The complexity of the pure 3-spin model at $\epsilon=0$ as a function of
+    The complexity of the 3-spin model at $\epsilon=0$ as a function of
     $a$ at several values of $\kappa$. The dashed line shows
     $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
   } \label{fig:complexity}
@@ -421,7 +420,7 @@ for the real problem, in a region where we expect this not to happen.
   \includegraphics{fig/desert.pdf}
   \caption{
     The minimum value of $a$ for which the complexity is positive as a function
-    of (real) energy $\epsilon$  for the pure 3-spin model at several values of
+    of (real) energy $\epsilon$  for the 3-spin model at several values of
     $\kappa$.
   } \label{fig:desert}
 \end{figure}
@@ -451,7 +450,7 @@ physical dynamics, are a problem we hope to address in future work.
 
   \caption{
     Energies at which states exist (green shaded region) and threshold energies
-    (black solid line) for the pure 3-spin model with
+    (black solid line) for the 3-spin model with
     $\kappa=\frac34e^{-i3\pi/4}$ and (a) $a=2$, (b) $a=1.325$, (c) $a=1.125$,
     and (d) $a=1$. No shaded region is shown in (d) because no states exist an
     any energy.
-- 
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