From 9fb7efbfafa6351f1af6cf5c46ccc104d6dcdf59 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 28 Jun 2024 17:13:16 +0200
Subject: Some wording and spacing changes to make a bad page spacing better.

---
 marginal.tex | 27 ++++++++++++---------------
 1 file changed, 12 insertions(+), 15 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 1fdaee4..bb4a96d 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1018,9 +1018,9 @@ fully connected couplings between the spheres, for which it is possible to also
 use configuration spaces involving spheres of different sizes
 \cite{Subag_2021_TAP, Subag_2023_TAP, Bates_2022_Crisanti-Sommers,
   Bates_2022_Free, Huang_2023_Strong, Huang_2023_Algorithmic,
-Huang_2024_Optimization}.
+Huang_2024_Optimization}. As far as we are aware, the deterministically coupled model has not been previously studied, except as a thought experiment in \cite{Kent-Dobias_2023_How}.
 
-The Lagrangian to be extremized to find stationary points and its gradient and Hessian are
+We again make use of the method of Lagrange multipliers to find stationary points on the constrained configuration space. The Lagrangian and its gradient and Hessian are
 \begin{align}
   &\begin{aligned}
     L(\mathbf x)&=H(\mathbf x)
@@ -1092,16 +1092,15 @@ different effective actions depending now on overlaps inside each of the two
 spheres and between the two spheres. The effective action for the traditional
 complexity of the multispherical spin glass is
 \begin{widetext}
-\begin{equation}
-  \begin{aligned}
+\begin{align}
     &\mathcal S_\mathrm{MSG}(\hat\beta,C^{11},R^{11},D^{11},G^{11},C^{22},R^{22},D^{22},G^{22},C^{12},R^{12},R^{21},D^{12},G^{12},G^{21}
-    \mid E,\omega_1,\omega_2)= \hat\beta(E-E_1-E_2-\epsilon c_d^{12})\\
+    \mid E,\omega_1,\omega_2)= \hat\beta(E-E_1-E_2-\epsilon c_d^{12}) \notag \\
     &
     +\mathcal S_\mathrm{SSG}(\hat\beta,C^{11},R^{11},D^{11},G^{11}\mid E_1,\omega_1)
     +\mathcal S_\mathrm{SSG}(\hat\beta,C^{22},R^{22},D^{22},G^{22}\mid E_2,\omega_2)
     +\lim_{n\to0}\frac1n\Bigg\{
       \epsilon\operatorname{Tr}(R^{12}+R^{21}+G^{12}+G^{21}-\hat\beta C^{12})
-    \\
+    \notag \\
     &\quad
       +\frac12\log\det\left(
     I-
@@ -1115,27 +1114,25 @@ complexity of the multispherical spin glass is
     \end{bmatrix}
     \right)
     -\log\det(I-(G^{11}G^{22})^{-1}G^{12}G^{21})\Bigg\}
-  \end{aligned}
-\end{equation}
+\end{align}
 which is the sum of two effective actions \eqref{eq:spherical.action} for the spherical spin glass
 associated with each individual sphere, and some coupling terms. The order
 parameters are defined the same as in the spherical spin glasses, but now with
 raised indices to indicate whether the vectors come from one or the other
 spherical subspace. The effective action for the eigenvalue-dependent part of
 the complexity is likewise given by
-\begin{equation} \label{eq:multispherical.marginal.action}
-  \begin{aligned}
-    &\mathcal U_\mathrm{MSG}(\hat q,\hat\lambda,Q^{11},Q^{22},Q^{12}\mid\beta,\lambda^*,\omega_1,\omega_2) \\
+\begin{align}
+    &\mathcal U_\mathrm{MSG}(\hat q,\hat\lambda,Q^{11},Q^{22},Q^{12}\mid\beta,\lambda^*,\omega_1,\omega_2) \notag \\
     &\quad=\lim_{m\to0}\bigg\{\sum_{\alpha=1}^m\left[\hat q^\alpha(Q^{11,\alpha\alpha}+Q^{22,\alpha\alpha}-1)-\beta(\omega_1Q^{11,\alpha\alpha}+\omega_2Q^{22,\alpha\alpha}-2\epsilon Q^{12,\alpha\alpha})\right]
-    +\hat\lambda(\omega_1Q^{11,11}+\omega_2Q^{22,11}-2\epsilon Q^{12,11}) \\
+    +\hat\lambda(\omega_1Q^{11,11}+\omega_2Q^{22,11}-2\epsilon Q^{12,11}) \notag \\
     &\qquad\qquad+\sum_{i=1,2}f_i''(1)\left[\beta^2\sum_{\alpha\gamma}^m(Q^{ii,\alpha\gamma})^2+2\beta\hat\lambda\sum_\alpha^m(Q^{ii,1\alpha})^2+\hat\lambda^2(Q^{ii,11})^2\right]
     +\frac12\log\det\begin{bmatrix}
       Q^{11}&Q^{12}\\
       Q^{12}&Q^{22}
     \end{bmatrix}
   \bigg\}
-  \end{aligned}
-\end{equation}
+\label{eq:multispherical.marginal.action}
+\end{align}
 \end{widetext}
 The new variables $\hat q^\alpha$ are Lagrange multipliers introduced
 to enforce the constraint that $Q^{11,\alpha\alpha}+Q^{22,\alpha\alpha}=1$.
@@ -1152,7 +1149,7 @@ diagonal not necessarily equal to 1, so
    \tilde q^{ij}_0 & q^{ij}_0 & q^{ij}_0 & \cdots & q^{ij}_d
  \end{bmatrix}
 \end{equation}
-This requires us to introduce two new order parameters per pair $(i,j)$. When
+This requires us to introduce two new order parameters per pair $(i,j)$. We also need two separate Lagrange multipliers $\hat q$ and $\hat{\tilde q}$ to enforce the tangent space normalization for the tilde and untilde replicas, respectively. When
 this ansatz is inserted into the expression \eqref{eq:multispherical.marginal.action} for the effective action and the
 limit of $m\to0$ is taken, we find
 \begin{widetext}
-- 
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