From 38dce748a67c967407b451c401c796a81f4f7425 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 24 Mar 2022 17:43:23 +0100
Subject: Breakthrough in p-spin continuation.

---
 stokes.tex | 93 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 84 insertions(+), 9 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index d4b0d38..1d3a8f3 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -771,9 +771,9 @@ stationary point of index $k$ has $k$ real eigenvectors and $D-k$ purely
 imaginary eigenvectors that contribute to its thimble. The matrix of
 eigenvectors can therefore be written $U=i^kO$ for an orthogonal matrix $O$,
 and with all eigenvectors canonically oriented $\det O=1$. We therefore have
-$\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvectors change: by a factor of $e^{-i\phi/2}$. Therefore, the contribution more generally is $(e^{-i\phi/2})^Di^k$. We therefore have, for real stationary points of a real action,
+$\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvectors change: by a factor of $e^{-i\phi/2}$. Therefore, the contribution more generally is $(e^{-i\phi/2})^Di^k$. We therefore have, for real stationary points of a real action before any Stokes points,
 \begin{equation}
-  Z_\sigma(\beta)\simeq\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}\prod_{\lambda_0>0}\lambda_0^{-\frac12}e^{-\beta\mathcal S(s_\sigma)}
+  Z_\sigma(\beta)\simeq\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta\mathcal S(s_\sigma)}
 \end{equation}
 
 \begin{eqnarray}
@@ -1071,17 +1071,17 @@ imaginary energy join.
 
 \begin{eqnarray}
   Z(\beta)
-  &=\int_{S^{N-1}}ds\,e^{-\beta H(s)}
+  &=\int_{S^{N-1}}ds\,e^{-\beta H_2(s)}
   =\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta H_2(s)} \\
   &\simeq\sum_{k=0}^D2i^k\left(\frac{2\pi}\beta\right)^{D/2}e^{-\beta H_2(s_\sigma)}\prod_{\lambda_0>0}\lambda_0^{-1/2} \\
   &=2\sum_{k=0}^D\exp\left\{
-    i\frac\pi2k+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon_k-\frac12\sum_{\ell\neq k}\log\frac12|\epsilon_k-\epsilon_\ell|
+    i\frac\pi2k+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon_k-\frac12\sum_{\ell\neq k}\log2|\epsilon_k-\epsilon_\ell|
   \right\}
 \end{eqnarray}
 \begin{eqnarray}
   Z(\beta)
   &=2\int d\epsilon\,\rho(\epsilon)\exp\left\{
-    i\frac\pi2k_\epsilon+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon-\frac D2\int d\epsilon'\,\rho(\epsilon')\log\frac12|\epsilon-\epsilon'|
+    i\frac\pi2k_\epsilon+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon-\frac D2\int d\epsilon'\,\rho(\epsilon')\log2|\epsilon-\epsilon'|
   \right\}
 \end{eqnarray}
 
@@ -1100,12 +1100,12 @@ The index as a function of energy level is given by the cumulative density funct
 \end{equation}
 Finally, the product over the singular values corresponding to descending directions gives
 \begin{equation}
-  \frac12\int d\epsilon'\,\rho(\epsilon')\log\frac12|\epsilon-\epsilon'|
-  =\log\frac{\epsilon_{\mathrm{th}}}2-\frac14+\frac12\left(\frac{\epsilon}{\epsilon_{\mathrm{th}}}\right)^2
+  \frac12\int d\epsilon'\,\rho(\epsilon')\log2|\epsilon-\epsilon'|
+  =-\frac14+\frac12\left(\frac{\epsilon}{\epsilon_{\mathrm{th}}}\right)^2
 \end{equation}
 for $\epsilon<\epsilon_{\mathrm{th}}$. Then
 \begin{equation}
-  \operatorname{Re}f=-\epsilon\operatorname{Re}\beta-\log\frac12+\frac14-\frac12\epsilon^2
+  \operatorname{Re}f=-\epsilon\operatorname{Re}\beta+\frac14-\frac12\epsilon^2
 \end{equation}
 \begin{equation}
   \operatorname{Im}f=-\epsilon\operatorname{Im}\beta+\frac12\left(
@@ -1442,11 +1442,86 @@ function of $\Delta$ and $\arg\delta$.
   }
 \end{figure}
 
+\subsection{Pure $p$-spin: is analytic continuation possible?}
+
+\begin{eqnarray}
+  Z(\beta)
+  &=\sum_{\sigma\in\Sigma_0}n_\sigma Z_\sigma(\beta) \\
+  &=\sum_{\sigma\in\Sigma_0}\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}
+    |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}
+    e^{-\beta\mathcal S(s_\sigma)} \\
+  &\simeq\sum_{k=0}^D\int d\epsilon\,\mathcal N_\mathrm{typ}(\epsilon,k)
+    \left(\frac{2\pi}\beta\right)^{D/2}i^k
+    \left(|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}\bigm|\mathcal S(s_\sigma)=N\epsilon,k_\sigma=k\right) e^{-\beta N\epsilon}
+\end{eqnarray}
+Following Derrida \cite{Derrida},
+\begin{equation}
+  \mathcal N_\mathrm{typ}(\epsilon,k)=\overline\mathcal N(\epsilon,k)+\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2}
+\end{equation}
+where $\eta$ is a random, sample-dependant number of order one. This gives two
+contributions to the partition function, but as we shall see it is the
+fluctuations in the number of stationary points at a given energy level that
+governs things at large $|\beta|$, not its total. This gives two terms to the typical partition function
+\begin{equation}
+  Z_\mathrm{typ}=Z_A+Z_B
+\end{equation}
+\begin{eqnarray}
+  Z_A
+  &\simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k)
+    \left(\frac{2\pi}\beta\right)^{D/2}i^k
+    |\det\operatorname{Hess}\mathcal S|^{-\frac12} e^{-\beta N\epsilon}
+    =\int d\epsilon\,e^{Nf_A(\epsilon)}
+    \\
+  Z_B
+  &\simeq\sum_{k=0}^D\int d\epsilon\,\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2}
+    \left(\frac{2\pi}\beta\right)^{D/2}i^k
+    |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon}
+    =\int d\epsilon\,\tilde\eta(\epsilon)e^{Nf_B(\epsilon)}
+\end{eqnarray}
+for
+\begin{eqnarray}
+  f_A
+  &=-\beta\epsilon+\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta
+    +i\frac\pi2P(\lambda<0\mid\epsilon) \\
+  f_B
+  &=-\beta\epsilon+\frac12\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta
+    +i\frac\pi2P(\lambda<0\mid\epsilon)
+\end{eqnarray}
+Each integral will be dominated by its value near the maximum of the real part of the exponential argument. Assuming that $\epsilon<\epsilon_\mathrm{th}$, this maximum occurs at
+\begin{equation}
+  0=-\operatorname{Re}\beta-\frac12\frac{3p-4}{p-1}\epsilon+\frac12\frac p{p-1}\sqrt{\epsilon^2-\epsilon_\mathrm{th}^2}
+\end{equation}
+\begin{equation}
+  0=-\operatorname{Re}\beta-\epsilon^2
+\end{equation}
+As with the 2-spin model, the integral over $\epsilon$ is oscillatory and can only be reliably evaluated with a saddle point when either the period of oscillation diverges \emph{or} when the maximum lies at a cusp. We therefore expect changes in behavior when $\epsilon=\epsilon_0$, the ground state energy. The temperature at which this happens is
+\begin{eqnarray}
+  \operatorname{Re}\beta_A&=-\frac12\frac{3p-4}{p-1}\epsilon_0+\frac12\frac p{p-1}\sqrt{\epsilon_0^2-\epsilon_\mathrm{th}^2}\\
+  \operatorname{Re}\beta_B&=-\epsilon_0
+\end{eqnarray}
+which for all $p\geq2$ has $\operatorname{Re}\beta_A>\operatorname{Re}\beta_B$.
+Therefore, the emergence of zeros in $Z_A$ does not lead to the emergence of
+zeros in the partition function as a whole, because $Z_B$ still produces a
+coherent result (despite the unknown constant factor $\eta(\epsilon_0)$). It is
+only at $\operatorname{Re}\beta_B=-\epsilon_0$ where both terms contributing to
+the partition function at large $N$ involve incoherent integrals near the
+maximum, and only here where the density of zeros is expected to become
+nonzero.
+
+In fact, in the limit of $|\beta|\to\infty$, $\operatorname{Re}\beta_B$ is
+precisely the transition found in \cite{Obuchi_2012_Partition-function} between
+phases with and without a density of zeros. This value is an underestimate for
+the transition for finite $|\beta|$, which likely results from the invalidity
+of our large-$\beta$ approximation. More of the phase diagram might be
+constructed by continuing the series for individual thimbles to higher powers
+in $\beta$, which would be equivalent to allowing non-constant terms in the
+Jacobian over the thimble.
+
+
 \section{The $p$-spin spherical models: numerics}
 
 To confirm the presence of Stokes lines under certain processes in the $p$-spin, we studied the problem numerically.
 
-
 \bibliographystyle{unsrt}
 \bibliography{stokes}
 
-- 
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