From f9a34f391dbae4e41d13b4ddde4bde6a8af8d93e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 2 Sep 2023 22:46:45 +0200
Subject: Work.

---
 marginal.tex | 98 +++++++++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 87 insertions(+), 11 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 59c9bc8..96c2a99 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -40,10 +40,11 @@
 An arbitrary function $g$ of the minimum eigenvalue of a matrix $A$ can be expressed as
 \begin{equation}
   g(\lambda_\textrm{min}(A))
-  =g\left(
-    \frac{x_\textrm{min}(A)^TAx_\textrm{min}(A)}N
-  \right)
-  =\frac12\lim_{\beta\to\infty}\int\frac{dx\,\delta(N-x^Tx)e^{\beta x^TAx}}{\int dx'\,\delta(N-x'^Tx')e^{\beta x'^TAx'}}g\left(\frac{x^TAx}N\right)
+  =\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)
+\end{equation}
+\begin{equation}
+  \lim_{\beta\to\infty}\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'}}
+  =d\mathbf x\,\frac12\left[\delta(\mathbf x_\mathrm{min}(A)-\mathbf x)+\delta(\mathbf x_\mathrm{min}(A)+\mathbf x)\right]
 \end{equation}
 The first equality makes use of the normalized eigenvector $x_\mathrm{min}(A)$
 associated with the minimum eigenvalue. By definition,
@@ -56,24 +57,99 @@ the limit of zero temperature, the measure will concentrate on the ground
 states of the model, which correspond with the eigenvectors $\pm x_\mathrm{min}$
 associated with the minimal eigenvalue $\lambda_\mathrm{min}$.
 
+
 \begin{equation}
-  d\mu_H(\mathbf s)=d\mathbf s\,\delta\big(\nabla H(\mathbf s)\big)\,\big|\det\operatorname{Hess}H(\mathbf s)\big|
+  H(\mathbf s)+\sum_{i=1}^r\omega_ig_i(\mathbf s)
 \end{equation}
+\begin{align}
+  \nabla H(\mathbf s,\pmb\omega)=\partial H(\mathbf s)+\sum_{i=1}^r\omega_i\partial g_i(\mathbf s)
+  &&
+  \operatorname{Hess}H(\mathbf s,\pmb\omega)=\partial\partial H(\mathbf s)+\sum_{i=1}^r\omega_i\partial\partial g_i(\mathbf s)
+\end{align}
+
 \begin{equation}
-  d\mu_H(\mathbf s\mid E)=d\mu_H(\mathbf s)\,\delta\big(NE-H(\mathbf s)\big)
+  d\mu_H(\mathbf s,\pmb\omega)=d\mathbf s\,d\pmb\omega\,\delta\big(\nabla H(\mathbf s,\pmb\omega)\big)\,\delta\big(\mathbf g(\mathbf s)\big)\,\big|\det\operatorname{Hess}H(\mathbf s,\pmb\omega)\big|
+\end{equation}
+\begin{equation}
+  d\mu_H(\mathbf s,\pmb\omega\mid E)=d\mu_H(\mathbf s,\pmb\omega)\,\delta\big(NE-H(\mathbf s)\big)
 \end{equation}
 
 \begin{equation}
   \begin{aligned}
-    \mathcal N_\text{marginal}(E)
-    &=\int d\mu_H(\mathbf s\mid E)\,\delta\big(\lambda_\mathrm{min}(\operatorname{Hess}H(\mathbf s))\big) \\
-    &=\frac12\lim_{\beta\to\infty}\lim_{m\to0}\int d\mu_H(\mathbf s\mid E)\int_{T_\mathbf s\Omega}\left(\prod_a^m dx_a\,\delta(N-x_a^Tx_a)e^{\beta x_a^TAx_a}\right)\,\delta\big(x_1^T\operatorname{Hess}H(\mathbf s)x_1\big)
+    &\mathcal N_\text{marginal}(E)
+    =\int d\mu_H(\mathbf s,\pmb\omega\mid E)\,\delta\big(N\lambda_\mathrm{min}(\operatorname{Hess}H(\mathbf s,\pmb\omega))\big) \\
+    &=\lim_{\beta\to\infty}\int d\mu_H(\mathbf s,\pmb\omega\mid E)
+    \frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)\delta(\mathbf x^T\partial\mathbf g(\mathbf s))e^{\beta\mathbf x^T\operatorname{Hess}H(\mathbf s,\pmb\omega)\mathbf x}}
+    {\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')\delta(\mathbf x'^T\partial\mathbf g(\mathbf s))e^{\beta\mathbf x'^T\operatorname{Hess}H(\mathbf s,\pmb\omega)\mathbf x'}}
+    \delta\big(\mathbf x^T\operatorname{Hess}H(\mathbf s,\pmb\omega)\mathbf x\big)
   \end{aligned}
 \end{equation}
+where the $\delta$-functions
+\begin{equation}
+  \delta(\mathbf x^T\partial\mathbf g(\mathbf s))
+  =\prod_{s=1}^r\delta(\mathbf x^T\partial g_i(\mathbf s))
+\end{equation}
+ensure that the integrals are constrained to the tangent space of the configuration manifold at the point $\mathbf s$.
 
 \begin{equation}
-  \beta^2f''(1)\sum A_{ab}^2+\hat x^2f''(1)A_{11}^2+\beta\hat xf''(1)\sum_a A_{1a}+\hat\beta^2f(C_{ab})+(2\hat\beta(R_{ab}-F_{ab})-D_{ab})f'(C_{a-b})+(R_{ab}^2-F_{ab}^2)f''(C_{ab})
-  +\log\det\begin{bmatrix}C&iR\\iR&D\end{bmatrix}-\log\det F
+  \begin{aligned}
+    &\Sigma_\text{marginal}(E)
+    =\frac1N\overline{\log\mathcal N_\text{marginal}(E)} \\
+    &=\lim_{\beta\to\infty}\lim_{n\to0}\frac\partial{\partial n}\int\prod_{a=1}^n\left[d\mu_H(\mathbf s_a,\pmb\omega_a\mid E)\lim_{m_a\to0}
+    \left(\prod_{b=1}^{m_a} d\mathbf x_a^b\,\delta(N-(\mathbf x_a^b)^T\mathbf x_a^b)\delta((\mathbf x_a^b)^T\partial\mathbf g(\mathbf s_a))e^{\beta(\mathbf x_a^b)^T\operatorname{Hess}H(\mathbf s_a,\pmb\omega_a)\mathbf x_a^b}\right)\,\delta\big((\mathbf x_a^1)^T\operatorname{Hess}H(\mathbf s_a,\pmb\omega_a)\mathbf x_a^1\big)\right]
+  \end{aligned}
+\end{equation}
+
+\section{Spherical model}
+
+\begin{align}
+  C_{ab}=\frac1N\mathbf s_a\cdot\mathbf s_b
+  &&
+  R_{ab}=-i\frac1N\mathbf s_a\cdot\hat{\mathbf s}_b
+  &&
+  D_{ab}=\frac1N\hat{\mathbf s}_a\cdot\hat{\mathbf s}_b
+  \\
+  A_{ab}^{cd}=\frac1N\mathbf x_a^c\cdot\mathbf x_b^d
+  &&
+  X^c_{ab}=\frac1N\mathbf s_a\cdot\mathbf x_b^c
+  &&
+  \hat X^c_{ab}=\frac1N\hat{\mathbf s}_a\cdot\mathbf x_b^c
+\end{align}
+
+\begin{equation}
+  \begin{aligned}
+    &\sum_{ab}^n\left[\beta\omega A_{aa}^{bb}+\hat x\omega A_{aa}^{11}+\beta^2f''(1)\sum_{cd}^m(A_{ab}^{cd})^2+\hat x^2f''(1)(A_{ab}^{11})^2+\beta\hat xf''(1)\sum_c^m A_{ab}^{1c}\right]\\
+    &+\hat\beta^2f(C_{ab})+(2\hat\beta(R_{ab}-F_{ab})-D_{ab})f'(C_{ab})+(R_{ab}^2-F_{ab}^2)f''(C_{ab})
+    +\log\det\begin{bmatrix}C&iR\\iR&D\end{bmatrix}-\log\det F
+  \end{aligned}
+\end{equation}
+
+$X^a$ is $n\times m_a$, and $A^{ab}$ is $m_a\times m_b$.
+\begin{equation}
+  \begin{bmatrix}
+    C&iR&X^1&\cdots&X^n \\
+    iR&D&i\hat X^1&\cdots&i\hat X^m\\
+    (X^1)^T&i(\hat X^1)^T&A^{11}&\cdots&A^{1n}\\
+    \vdots&\vdots&\vdots&\ddots&\vdots\\
+    (X^n)^T&i(\hat X^n)^T&A^{n1}&\cdots&A^{nn}
+  \end{bmatrix}
+\end{equation}
+$X_{ab}^c$ will be nonzero if the lowest eigenvector of the hessian at the
+point $\mathbf s_c$ are correlated with the direction of the point $\mathbf
+s_a$. Since the eigenvector problem is always expected to be replica symmetric,
+we expect no $b$-dependence of this matrix. $A^{aa}$ is the usual
+replica-symmetric overlap matrix of the spherical two-spin problem. $A^{ab}$
+describes overlaps between eigenvectors at different stationary points and should be a constant $m_a\times m_b$ matrix.
+
+We will discuss at the end of this paper when these order parameters can be expected to be nonzero, but in this problem all of the $X$s, $\hat X$s, and $A^{ab}$ for $a\neq b$ are zero.
+
+\begin{equation}
+  \begin{aligned}
+    &\sum_a^n m_a\beta\omega+m_1\hat x\omega+\sum_{a}^n\left[m_a\beta^2f''(1)(1+(m_a-1)a_0^2)+\hat x^2f''(1)m_a+\beta\hat xf''(1)(1+(m_a-1)a_0)\right]\\
+    &+\hat\beta^2f(C_{ab})+(2\hat\beta(R_{ab}-F_{ab})-D_{ab})f'(C_{ab})+(R_{ab}^2-F_{ab}^2)f''(C_{ab})
+    +\log\det\begin{bmatrix}C&iR\\iR&D\end{bmatrix}-\log\det F
+    +mn\log(1-a_0)+mn\frac{a_0}{1-a_0}
+  \end{aligned}
 \end{equation}
 
 \section{Superfield formalism}
-- 
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