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\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.

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