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\documentclass[fleqn,a4paper]{article}
\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
\usepackage[T1]{fontenc} % vector fonts plz
\usepackage{fullpage,amsmath,amssymb,latexsym,graphicx}
\usepackage{newtxtext,newtxmath} % Times for PR
\usepackage{appendix}
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]{biblatex}
\usepackage{anyfontsize,authblk}
\usepackage{tikz}
\addbibresource{marginal.bib}
\begin{document}
\title{
None yet
}
\author{Jaron Kent-Dobias}
\affil{Istituto Nazionale di Fisica Nucleare, Sezione di Roma I}
%\maketitle
%\begin{abstract}
%\end{abstract}
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)
\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,
$x_\mathrm{min}(A)^TAx_\mathrm{min}(A)=x_\mathrm{min}(A)^Tx_\mathrm{min}(A)\lambda_\mathrm{min}(A)=N\lambda_\mathrm{min}(A)$
assuming the normalization is $\|x_\mathrm{min}(A)\|^2=N$. The second equality
extends a technique first introduced in \cite{Ikeda_2023_Bose-Einstein-like}
and used in \cite{me}. A Boltzmann distribution is introduced 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 eigenvectors $\pm x_\mathrm{min}$
associated with the minimal eigenvalue $\lambda_\mathrm{min}$.
\begin{equation}
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,\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,\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}
\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 and most isotropic problems 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]\\
&+\mathcal S(\hat\beta,C,R,D) -2\hat\beta F_{ab}f'(C_{ab})-F_{ab}^2f''(C_{ab})
-\log\det F
+mn\log(1-a_0)+mn\frac{a_0}{1-a_0}
\end{aligned}
\end{equation}
\section{Superfield formalism}
\begin{equation}
\pmb\phi_a(1)=\pmb\sigma_a+\bar\theta(1)\pmb\eta_a+\bar{\pmb\eta}_a\theta(1)+\hat{\pmb\sigma}_a\bar\theta(1)\theta(1)
\end{equation}
\begin{equation}
\pmb\xi_b(1)=\pmb\sigma_1+\mathbf x_b\bar\vartheta(1)\theta(1)+\mathbf x_b\bar\theta(1)\vartheta(1)
\end{equation}
\begin{equation}
\int d\theta\,d\bar\theta\,\left[
(1+\hat\beta\bar\theta\theta)H(\pmb\phi)
+\int d\vartheta\,d\bar\vartheta\,H(\pmb\xi)
\right]
=\hat{\pmb\sigma}^T\partial H(\pmb\sigma)
+\pmb\eta^T\partial\partial H(\pmb\sigma)\pmb\eta
+\beta\mathbf x^T\partial\partial H(\pmb\sigma)\mathbf x
+\hat\beta H(\pmb\sigma)
\end{equation}
\begin{equation}
\int d1\,d2\,(1+\hat\beta\bar\theta(1)\theta(1))(1+\hat\beta\bar\theta(2)\theta(2))
f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right)
+\int d1\,(1+\hat\beta\bar\theta(1)\theta(1))f\left(\frac{\pmb\phi_a(1)\cdot\pmb\xi_b(2)}N\right)
\end{equation}
\section{Twin spherical model}
$\Omega=S^{N-1}\times S^{N-1}$
\begin{equation}
H(\pmb\sigma)=H_1(\pmb\sigma^{(1)})+H_2(\pmb\sigma^{(2)})+\epsilon\pmb\sigma^{(1)}\cdot\pmb\sigma^{(2)}
\end{equation}
\begin{equation}
\overline{H_s(\pmb\sigma_1)H_s(\pmb\sigma_2)}
=Nf_s\left(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\right)
\end{equation}
\section{Multi-species spherical model}
We consider models whose configuration space consists of the product of $r$
spheres, each with its own dimension $N_s$, or
$\Omega=S^{N_1-1}\times\cdots\times S^{N_r-1}$. Coordinates on this space we
will typically denote
$\pmb\sigma=(\pmb\sigma^{(1)},\ldots,\pmb\sigma^{(r)})\in\Omega$, with
$\pmb\sigma^{(s)}\in S^{N_s-1}$ denoting the subcomponent restricted to a
specific subsphere. The model can be thought of as consisting of centered
random functions $H:\Omega\to\mathbb R$ with covariance
\begin{equation}
\overline{
H(\pmb\sigma_1)H(\pmb\sigma_2)
}
=f\left(
\frac{\pmb\sigma^{(1)}_1\cdot\pmb\sigma^{(1)}_2}{N_1},
\ldots,
\frac{\pmb\sigma^{(r)}_1\cdot\pmb\sigma^{(r)}_2}{N_r}
\right)
\end{equation}
where $f:[-1,1]^r\to\mathbb R$ is an $r$-component function of the overlaps that defines the model.
\printbibliography
\end{document}
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