From 4aae138676907599c1fbfe698da2bb7861d4f12e Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Aug 2023 18:49:52 +0200 Subject: Initial commit. --- .gitignore | 14 ++++++++ marginal.bib | 14 ++++++++ marginal.tex | 111 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 139 insertions(+) create mode 100644 .gitignore create mode 100644 marginal.bib create mode 100644 marginal.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..9a36fc7 --- /dev/null +++ b/.gitignore @@ -0,0 +1,14 @@ +*.aux +*.fdb_latexmk +*.fls +*.log +/*.pdf +*.synctex.gz +*.bbl +*.blg +*.out +*.bcf +*.run.xml +*.synctex(busy) +*.toc +*Notes.bib diff --git a/marginal.bib b/marginal.bib new file mode 100644 index 0000000..9ea0fce --- /dev/null +++ b/marginal.bib @@ -0,0 +1,14 @@ +@article{Ikeda_2023_Bose-Einstein-like, + author = {Ikeda, Harukuni}, + title = {{Bose}--{Einstein}-like condensation of deformed random matrix: a replica approach}, + journal = {Journal of Statistical Mechanics: Theory and Experiment}, + publisher = {IOP Publishing}, + year = {2023}, + month = {2}, + number = {2}, + volume = {2023}, + pages = {023302}, + url = {https://doi.org/10.1088%2F1742-5468%2Facb7d6}, + doi = {10.1088/1742-5468/acb7d6} +} + diff --git a/marginal.tex b/marginal.tex new file mode 100644 index 0000000..fb212db --- /dev/null +++ b/marginal.tex @@ -0,0 +1,111 @@ +\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