From 4aae138676907599c1fbfe698da2bb7861d4f12e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Aug 2023 18:49:52 +0200
Subject: Initial commit.

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 marginal.tex | 111 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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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}
+\usepackage[dvipsnames]{xcolor}
+\usepackage[
+  colorlinks=true,
+  urlcolor=MidnightBlue,
+  citecolor=MidnightBlue,
+  filecolor=MidnightBlue,
+  linkcolor=MidnightBlue
+]{hyperref} % ref and cite links with pretty colors
+\usepackage[
+  style=phys,
+  eprint=true,
+  maxnames = 100
+]{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))
+  =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)
+\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}
+  d\mu_H(\mathbf s)=d\mathbf s\,\delta\big(\nabla H(\mathbf s)\big)\,\big|\det\operatorname{Hess}H(\mathbf s)\big|
+\end{equation}
+\begin{equation}
+  d\mu_H(\mathbf s\mid E)=d\mu_H(\mathbf s)\,\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)
+  \end{aligned}
+\end{equation}
+
+\section{Superfield formalism}
+
+\begin{equation}
+  \pmb\phi=\pmb\sigma+\bar\theta\pmb\eta+\bar{\pmb\eta}\theta+\hat{\pmb\sigma}\bar\theta\theta+\mathbf x\bar\vartheta\theta+\mathbf x\bar\theta\vartheta
+\end{equation}
+\begin{equation}
+  \int d\theta\,d\bar\theta\,d\vartheta\,d\bar\vartheta\,(\bar\vartheta\vartheta+\beta+\hat\beta\bar\vartheta\vartheta\bar\theta\theta)H(\pmb\phi)
+  =\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}
+
+\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}
-- 
cgit v1.2.3-70-g09d2

