From 0368a5d84845a2de6050d206f456c648234c5761 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 21 Feb 2022 15:19:48 +0100
Subject: Work on the 2-spin section.

---
 stokes.tex | 72 ++++++++++++++++++++++++++++++++++++++++++++++++++++----------
 1 file changed, 61 insertions(+), 11 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 41e21b2..84757b9 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1052,22 +1052,72 @@ imaginary energy join.
   Z(\beta)
   &=\int_{S^{N-1}}ds\,e^{-\beta H(s)}
   =\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta H_2(s)} \\
-  &\simeq\sum_{k=0}^D2n_ki^k\left(\frac{2\pi}\beta\right)^{D/2}e^{-\beta H_2(s_\sigma)}\prod_{\lambda_0>0}\lambda_0^{-1/2} \\
-  &=\sum_{k=0}^D2n_ki^k\left(\frac{2\pi}\beta\right)^{D/2}e^{-N\beta\epsilon_k}\prod_{\ell\neq k}\frac12|\epsilon_k-\epsilon_\ell|
+  &\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|
+  \right\}
 \end{eqnarray}
-
 \begin{eqnarray}
   Z(\beta)
-  &=\int_{S^{N-1}}ds\,e^{-\beta H(s)}
-  =\int_{\mathbb R^N}dx\,\delta(x^2-N)e^{-\beta H(x)} \\
-  &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12\beta x^TJx-\lambda(x^Tx-N)} \\
-  &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12x^T(\beta J+\lambda I)x+\lambda N} \\
-  &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\det(\beta J+\lambda I)}}e^{\lambda N} \\
-  &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\prod_i(\beta\lambda_i+\lambda)}}e^{\lambda N} \\
-  &=(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac12\sum_i\log(\beta\lambda_i+\lambda)} \\
-  &\simeq(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac N2\int d\lambda'\,\rho(\lambda')\log(\beta\lambda'+\lambda)} \\
+  &=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'|
+  \right\}
 \end{eqnarray}
 
+Since the $J$ of the 2-spin model is a symmetric real matrix with variance
+$1/N$, its eigenvalues are distributed by a semicircle distribution of radius 2,
+and therefore the energies $\epsilon$ are distributed by a semicircle
+distribution of radius $\epsilon_{\mathrm{th}}=1$, with
+\begin{equation}
+  \rho(\epsilon\mid\epsilon_{\mathrm{th}})=\frac2{\pi\epsilon_{\mathrm{th}}^2}\sqrt{\epsilon_{\mathrm{th}}^2-\epsilon^2}
+\end{equation}
+The index as a function of energy level is given by the cumulative density function
+\begin{equation}
+  k_\epsilon=N\int_{-\infty}^\epsilon d\epsilon'\,\rho(\epsilon')=\frac N\pi\left(
+    \frac{\epsilon}{\epsilon_{\mathrm{th}}^2}\sqrt{\epsilon_{\mathrm{th}}^2-\epsilon^2}+2\tan^{-1}\frac{\epsilon_{\mathrm{th}}+\epsilon}{\sqrt{\epsilon_{\mathrm{th}}^2-\epsilon^2}}
+  \right)
+\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
+\end{equation}
+for $\epsilon<\epsilon_{\mathrm{th}}$. Then
+\begin{equation}
+  \operatorname{Re}f=-\epsilon\operatorname{Re}\beta-\log\frac12+\frac14-\frac12\epsilon^2
+\end{equation}
+\begin{equation}
+  \operatorname{Im}f=-\epsilon\operatorname{Im}\beta+\frac12\left(
+    \epsilon\sqrt{1-\epsilon^2}+2\tan^{-1}\frac{1+\epsilon}{\sqrt{1-\epsilon^2}}
+  \right)
+\end{equation}
+The value of the integral will be dominated by the contribution near the
+maximum of the real part of $f$, which is
+\begin{equation}
+  \epsilon_{\mathrm{max}}=\left\{\matrix{-\operatorname{Re}\beta & \operatorname{Re}\beta\leq1\cr -1 & \mathrm{otherwise}}\right.
+\end{equation}
+For $\operatorname{Re}\beta>1$, the maximum is concentrated in the ground state
+and the real part of $f$ comes to a cusp, meaning that the oscillations do not
+interfere in taking the saddle point. Once this line is crossed and the maximum
+enters the bulk of the spectrum, one expects to find cancellations caused by
+the incoherent contributions of thimbles with nearby energies to
+$\epsilon_{\mathrm{max}}$.
+
+On the other hand, there is another point where the thimble sum becomes
+coherent. This is when the oscillation frequency near the maximum energy goes
+to zero. This happens for
+\begin{equation}
+  0
+  =\frac{\partial}{\partial\epsilon}\operatorname{Im}f\Big|_{\epsilon=\epsilon_{\mathrm{max}}}
+  =-\operatorname{Im}\beta+\sqrt{1-\epsilon_{\mathrm{max}}^2}
+  =-\operatorname{Im}\beta+\sqrt{1-(\operatorname{Re}\beta)^2}
+\end{equation}
+or for $|\beta|=1$. Here the sum of contributions from thimbles near the
+maximum again becomes coherent. These conditions correspond precisely to those
+found for the density of zeros in the 2-spin model found previously
+\cite{Obuchi_2012_Partition-function, Takahashi_2013_Zeros}.
+
+
 \subsection{Pure \textit{p}-spin: where are the saddles?}
 
 Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also
-- 
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