From 63eacd9d14ded6cc306c11e759398071bef3ee76 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 11 May 2024 22:25:11 +0200
Subject: Started translation to maximum eigenvalue.

---
 marginal.tex | 25 ++++++++++++++-----------
 1 file changed, 14 insertions(+), 11 deletions(-)

diff --git a/marginal.tex b/marginal.tex
index f19c096..ec28568 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -86,19 +86,23 @@ more useful.
 
 
 
-An arbitrary function $g$ of the minimum eigenvalue of a matrix $A$ can be expressed as
+An arbitrary function $g$ of the minimum eigenvalue of a matrix $A$ can be
+expressed as
 \begin{equation}
-  g(\lambda_\textrm{min}(A))
-  =\lim_{\beta\to\infty}\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)e^{-\beta\mathbf x^TA\mathbf x}}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')e^{-\beta\mathbf x'^TA\mathbf x'}}g\left(\frac{\mathbf x^TA\mathbf x}N\right)
+  g(\lambda_\textrm{max}(A))
+  =\lim_{\beta\to\infty}\int
+    \frac{d\mathbf s\,\delta(N-\mathbf s^T\mathbf s)e^{\beta\mathbf s^TA\mathbf s}}
+      {\int d\mathbf s'\,\delta(N-\mathbf s'^T\mathbf s')e^{\beta\mathbf s'^TA\mathbf s'}}
+    g\left(\frac{\mathbf s^TA\mathbf s}N\right)
 \end{equation}
 Assuming
 \begin{equation}
   \begin{aligned}
-    &\lim_{\beta\to\infty}\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)e^{-\beta\mathbf x^TA\mathbf x}}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')e^{-\beta\mathbf x'^TA\mathbf x'}}g\left(\frac{\mathbf x^TA\mathbf x}N\right) \\
-    &=\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x)}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x')}g\left(\frac{\mathbf x^TA\mathbf x}N\right) \\
-    &=g(\lambda_\mathrm{min}(A))
-    \frac{\int d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x)}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x')} \\
-    &=g(\lambda_\mathrm{min}(A))
+    &\lim_{\beta\to\infty}\int\frac{d\mathbf s\,\delta(N-\mathbf s^T\mathbf s)e^{\beta\mathbf s^TA\mathbf s}}{\int d\mathbf s'\,\delta(N-\mathbf s'^T\mathbf s')e^{\beta\mathbf s'^TA\mathbf s'}}g\left(\frac{\mathbf s^TA\mathbf s}N\right) \\
+    &=\int\frac{d\mathbf s\,\delta(N-\mathbf s^T\mathbf s)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{max}(A)I)}(\mathbf s)}{\int d\mathbf s'\,\delta(N-\mathbf s'^T\mathbf s')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{max}(A)I)}(\mathbf s')}g\left(\frac{\mathbf s^TA\mathbf s}N\right) \\
+    &=g(\lambda_\mathrm{max}(A))
+    \frac{\int d\mathbf s\,\delta(N-\mathbf s^T\mathbf s)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{max}(A)I)}(\mathbf s)}{\int d\mathbf s'\,\delta(N-\mathbf s'^T\mathbf s')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{max}(A)I)}(\mathbf s')} \\
+    &=g(\lambda_\mathrm{max}(A))
   \end{aligned}
 \end{equation}
 The first relation extends a technique first introduced in
@@ -108,9 +112,8 @@ over a spherical model whose Hamiltonian is quadratic with interaction matrix
 given by $A$. In the limit of zero temperature, the measure will concentrate on
 the ground states of the model, which correspond with the eigenspace of $A$
 associated with its minimum eigenvalue $\lambda_\mathrm{min}$. The second
-relation uses the fact that, once restricted to the sphere $\mathbf x^T\mathbf
-x=N$ and the minimum eigenspace, $\mathbf x^TA\mathbf
-x=N\lambda_\mathrm{min}(A)$.
+relation uses the fact that, once restricted to the sphere $\mathbf s^T\mathbf
+s=N$ and the minimum eigenspace, $\mathbf s^TA\mathbf s=N\lambda_\mathrm{min}(A)$.
 
 The relationship is formal, but we can make use of the fact that the integral
 expression with a Gibbs distribution can be manipulated with replica
-- 
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