From 9e7a47e1e28bbacc4a05cc1328ac073e867fafb3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 10 Oct 2023 17:11:13 +0200
Subject: Added details for taking the saddle point in the spherical models.

---
 marginal.tex | 87 +++++++++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 78 insertions(+), 9 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index e9dc5a8..f06bc04 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -73,7 +73,7 @@ Taking the average over $B$, we have
   e^{NG_\sigma(\omega)}
   =\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\prod_{a=1}^n\left[d\mathbf x_a\,\delta(N-\mathbf x_a^T\mathbf x_a)\right]
   \exp\left\{-Nn\beta\omega+N\lambda\omega+\frac{\sigma^2}{N}\left[\beta^2\sum_{ab}^n(\mathbf x_a^T\mathbf x_b)^2
-    +2\beta\lambda\sum_a^n(\mathbf x_a^T\mathbf x_1)^2
+    -2\beta\lambda\sum_a^n(\mathbf x_a^T\mathbf x_1)^2
     +\lambda^2N^2
   \right]\right\}
 \end{equation}
@@ -83,7 +83,7 @@ We make the Hubbard--Stratonovich transformation to the matrix field $Q_{ab}=\fr
   =\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\,dQ\,
   \exp N\left\{
     -n\beta\omega+\lambda\omega+\sigma^2\left[\beta^2\sum_{ab}^nQ_{ab}^2
-      +2\beta\lambda\sum_a^nQ_{1a}^2
+      +-\beta\lambda\sum_a^nQ_{1a}^2
     +\lambda^2
   \right]+\frac12\log\det Q\right\}
 \end{equation}
@@ -109,17 +109,58 @@ Inserting these expressions and taking the limit of $n$ to zero, we find
 \begin{equation}
   \mathcal S(q_0,\tilde q_0,\lambda)=\lambda\omega+\sigma^2\left[
     2\beta^2(q_0^2-\tilde q_0^2)-2\beta\lambda(1-\tilde q_0^2)+\lambda^2
-  \right]-\log(1-q_0)+\frac12\log(1-2q_0+\tilde q_1^2)
+  \right]-\log(1-q_0)+\frac12\log(1-2q_0+\tilde q_0^2)
 \end{equation}
 The integral is then given by its value at the stationary point of this
-expression with respect to its three arguments. Making the extremization and taking the limit of $\beta$ to infinity, we find
+expression with respect to its three arguments.
+The extremal conditions are
+\begin{align}
+  0&=\frac{\partial\mathcal S}{\partial q_0}
+  =\frac1{1-q_0}-\frac1{1-2q_0+\tilde q_0^2}+4\beta^2\sigma^2q_0 \\
+  0&=\frac{\partial\mathcal S}{\partial \tilde q_0}
+  =\frac{\tilde q_0}{1-2q_0+\tilde q_0^2}-4\sigma^2(\beta^2-\beta\lambda)\tilde q_0 \\
+  0&=\frac{\partial\mathcal S}{\partial\lambda}
+  =\omega+2\sigma^2\big(\lambda-\beta(1-\tilde q_0^2)\big)
+\end{align}
+We expect the overlaps to concentrate on one as $\beta$ goes to infinity. We therefore take
+\begin{align}
+  q_0=1-y\beta^{-1}-z\beta^{-2}+O(\beta^{-3})
+  &&
+  \tilde q_0=1-\tilde y\beta^{-1}-\tilde z\beta^{-2}+O(\beta^{-3})
+\end{align}
+The first equations expanded in $\beta$ give
+\begin{align}
+  &0=4\sigma^2\beta^2+\bigg(\frac1{y}-\frac12\frac1{y-\tilde y}-4y\sigma^2\bigg)\beta+O(\beta^0) \\
+  &0=-4\sigma^2\beta^2+\bigg(\frac12\frac1{y-\tilde y}+4\sigma^2(\lambda+\tilde y)\bigg)\beta+O(\beta^0)
+\end{align}
+One cannot satisfy this equation order-by-order in $\beta$. However, a solution
+suggests itself: the expansion is singular when $\tilde y=y$. Making this
+identification, we find instead
+\begin{align}
+  &0=\left(4\sigma^2-\frac1{y^2+2(z-\tilde z)}\right)\beta^2+\left(\frac1y+\frac{2y\tilde z}{(y^2+2(z-\tilde z))^2}-4\sigma^2y\right)\beta+O(\beta^0) \\
+  &0=\left(-4\sigma^2+\frac1{y^2+2(z-\tilde z)}\right)\beta^2+\left(-\frac y{y^2+2(z-\tilde z)}-\frac{2y\tilde z}{(y^2+2(z-\tilde z))^2}+4\sigma^2(y+\lambda)\right)\beta+O(\beta^0)
+\end{align}
+Finally, expanding the equation for $\lambda$ to lowest order, we have
+\begin{equation}
+  0=\omega+2\sigma^2(\lambda-2y)+O(\beta^{-1})
+\end{equation}
+Simultaneously solving these five equations stemming from the coefficients of $\beta$ for $y$, $z$, $\tilde z$, and $\lambda$, we have
+\begin{align}
+  \lambda=-\frac1\sigma\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
+  &&
+  y=\frac1{2\sigma}\left(\frac{\omega}{2\sigma}-\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}\right)
+  \\
+  z=\frac1{2\sigma^2}\left(1-\frac{\omega^2}{(2\sigma)^2}\right)
+  &&
+  \tilde z=\frac1{4\sigma^2}\left(1-\frac{\omega}{2\sigma}\left(\frac\omega{2\sigma}+\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}\right)\right)
+\end{align}
+Inserting this solution into $\mathcal S$ and taking the limit of $\beta$ to zero, we find
 \begin{equation}
   G_\sigma(\omega)=-\frac{\omega}{2\sigma}\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
-  -\log\left[
-    \frac{\omega}{2\sigma}-\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
+  +\log\left[
+    \frac{\omega}{2\sigma}+\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
   \right]
 \end{equation}
-where the branch of the square roots is the same as the sign of $\omega-2\sigma$.
 This function is plotted in Fig. For $\omega<2\sigma$ $G_\sigma(\omega)$ has an
 imaginary part, which makes any additional integral over $\omega$ highly
 oscillatory. This indicates that the existence of a marginal minimum for this
@@ -260,12 +301,40 @@ $\Omega=S^{N-1}\times S^{N-1}$
 \end{equation}
 
 \begin{equation}
-  \mathcal S_1(C^{11},R^{11},D^{11},\hat\beta,\omega_1)+\mathcal S_2(C^{22},R^{22},D^{22},\hat\beta,\omega_2)
-  -\epsilon r_{12}-\epsilon r_{21}-\omega_1r^{11}_d-\omega_2r^{22}_d+\hat\beta E
+  \mathcal S(C,R,D,W,\hat\beta,\omega)
+  =\frac12\frac1n
+    \sum_{ab}\left(
+      \hat\beta^2f(C_{ab})+(2\hat\beta R_{ab}-D_{ab})f'(C_{ab})+(R_{ab}^2-W_{ab}^2)f''(C_{ab})
+    \right)
+\end{equation}
+
+\begin{equation}
+  \mathcal S(C^{11},R^{11},D^{11},W^{11},\hat\beta)+\mathcal S(C^{22},R^{22},D^{22},W^{22},\hat\beta)
+  -\epsilon(r_{12}+r_{21})-\omega_1(r^{11}_d-w^{11}_d)-\omega_2(r^{22}_d-w^{22}_d)+\hat\beta E
   +\frac12\log\det\begin{bmatrix}C^{11}&iR^{11}\\iR^{11}&D^{11}\end{bmatrix}
   +\frac12\log\det\left(
   \begin{bmatrix}C^{22}-q_{12}^2C^{11}&iR^{22}\\iR^{22}&D^{22}\end{bmatrix}
   \right)
+  -\log\det(W^{11}W^{22}+W^{12}W^{21})
+\end{equation}
+
+\begin{equation}
+  \begin{aligned}
+    &\sum_a^n\left[\hat q_a(Q^{11}_{aa}+Q^{22}_{aa}-1)-\beta(\omega_1Q^{11}_{aa}+\omega_2Q^{22}_{aa}+2\epsilon Q^{12}_{aa})\right]
+    +\lambda(\omega_1Q^{11}_{11}+\omega_2Q^{22}_{11}+2\epsilon Q^{12}_{11}) \\
+    &+\sum_{i=1,2}f_i''(1)\left[\beta^2\sum_{ab}^n(Q^{ii}_{ab})^2-2\beta\lambda\sum_a^n(Q^{ii}_{1a})^2+\lambda^2(Q^{ii}_{11})^2\right]
+    +\frac12\log\det\begin{bmatrix}
+      Q^{11}&Q^{12}\\
+      Q^{12}&Q^{22}
+    \end{bmatrix}
+  \end{aligned}
+\end{equation}
+\begin{equation}
+  \log\det\begin{bmatrix}
+    Q^{11}&Q^{12}\\
+    Q^{12}&Q^{22}
+  \end{bmatrix}
+  +\log\det(Q^{11}Q^{22}-Q^{12}Q^{12})
 \end{equation}
 
 \section{Multi-species spherical model}
-- 
cgit v1.2.3-70-g09d2

