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authorJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-10 17:13:40 +0100
committerJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-10 17:13:40 +0100
commitd0945d932b815a7eeab00723d4b76d32d037216f (patch)
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parent50a9abd97a9a7fbe88a499a9c300bf4a42864510 (diff)
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More small changes.
-rw-r--r--bezout.tex21
1 files 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.