path: root/marginal.tex

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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))
  =\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 $\lambda_\mathrm{min}$.

Consider a matrix $A=B+\omega I$ for $B$ a GOE matrix with entries whose variance is $\sigma^2/N$. As an example, we compute
\begin{equation}
  e^{NG_\sigma(\omega)}=\overline{\lim_{\beta\to\infty}\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)e^{-\beta\mathbf x^T(B+\omega I)\mathbf x}}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')e^{-\beta\mathbf x'^T(B+\omega I)\mathbf x'}}\,\delta\big(\mathbf x^T(B+\omega I)\mathbf x\big)}
\end{equation}
where the overline is the average over $B$. Using replicas to treat the
denominator and transforming the $\delta$-function to its Fourier
representation, we have
\begin{equation}
  e^{NG_\sigma(\omega)}=\overline{\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\prod_{a=1}^n\left[d\mathbf x_a\,\delta(N-\mathbf x_a^T\mathbf x_a)\right]
  \exp\left\{-\beta\sum_{a=1}^n\mathbf x_a^T(B+\omega I)\mathbf x_a+\lambda\mathbf x_1^T(B+\omega I)\mathbf x_1\right\}}
\end{equation}
Taking the average over $B$, we have
\begin{equation}
  e^{NG_\sigma(\omega)}
  =\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\prod_{a=1}^n\left[d\mathbf x_a\,\delta(N-\mathbf x_a^T\mathbf x_a)\right]
  \exp\left\{-Nn\beta\omega+N\lambda\omega+\frac{\sigma^2}{N}\left[\beta^2\sum_{ab}^n(\mathbf x_a^T\mathbf x_b)^2
    -2\beta\lambda\sum_a^n(\mathbf x_a^T\mathbf x_1)^2
    +\lambda^2N^2
  \right]\right\}
\end{equation}
We make the Hubbard--Stratonovich transformation to the matrix field $Q_{ab}=\frac1N\mathbf x_a^T\mathbf x_b$. This gives
\begin{equation}
  e^{NG_\sigma(\omega)}
  =\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\,dQ\,
  \exp N\left\{
    -n\beta\omega+\lambda\omega+\sigma^2\left[\beta^2\sum_{ab}^nQ_{ab}^2
      +-\beta\lambda\sum_a^nQ_{1a}^2
    +\lambda^2
  \right]+\frac12\log\det Q\right\}
\end{equation}
where $Q_{aa}=1$ because of the spherical constraint. We can evaluate this
integral using the saddle point method. We make a replica symmetric ansatz for
$Q$, because this is a 2-spin model, but with the first row singled out because
of its unique coupling with $\lambda$. This gives
\begin{equation}
  Q=\begin{bmatrix}
    1&\tilde q_0&\tilde q_0&\cdots&\tilde q_0\\
    \tilde q_0&1&q_0&\cdots&q_0\\
    \tilde q_0&q_0&1&\ddots&q_0\\
    \vdots&\vdots&\ddots&\ddots&\vdots\\
    \tilde q_0&q_0&q_0&\cdots&q_0
  \end{bmatrix}
\end{equation}
with $\sum_{ab}Q_{ab}^2=n+2(n-1)\tilde q_0^2+(n-1)(n-2)q_0^2$, $\sum_aQ_{1a}^2=1+(n-1)\tilde q_0^2$,
and
\begin{equation}
  \log\det Q=(n-2)\log(1-q_0)+\log(1+(n-2)q_0-(n-1)\tilde q_0^2)
\end{equation}
Inserting these expressions and taking the limit of $n$ to zero, we find
\begin{equation}
  \mathcal S(q_0,\tilde q_0,\lambda)=\lambda\omega+\sigma^2\left[
    2\beta^2(q_0^2-\tilde q_0^2)-2\beta\lambda(1-\tilde q_0^2)+\lambda^2
  \right]-\log(1-q_0)+\frac12\log(1-2q_0+\tilde q_0^2)
\end{equation}
The integral is then given by its value at the stationary point of this
expression with respect to its three arguments.
The extremal conditions are
\begin{align}
  0&=\frac{\partial\mathcal S}{\partial q_0}
  =\frac1{1-q_0}-\frac1{1-2q_0+\tilde q_0^2}+4\beta^2\sigma^2q_0 \\
  0&=\frac{\partial\mathcal S}{\partial \tilde q_0}
  =\frac{\tilde q_0}{1-2q_0+\tilde q_0^2}-4\sigma^2(\beta^2-\beta\lambda)\tilde q_0 \\
  0&=\frac{\partial\mathcal S}{\partial\lambda}
  =\omega+2\sigma^2\big(\lambda-\beta(1-\tilde q_0^2)\big)
\end{align}
We expect the overlaps to concentrate on one as $\beta$ goes to infinity. We therefore take
\begin{align}
  q_0=1-y\beta^{-1}-z\beta^{-2}+O(\beta^{-3})
  &&
  \tilde q_0=1-\tilde y\beta^{-1}-\tilde z\beta^{-2}+O(\beta^{-3})
\end{align}
The first equations expanded in $\beta$ give
\begin{align}
  &0=4\sigma^2\beta^2+\bigg(\frac1{y}-\frac12\frac1{y-\tilde y}-4y\sigma^2\bigg)\beta+O(\beta^0) \\
  &0=-4\sigma^2\beta^2+\bigg(\frac12\frac1{y-\tilde y}+4\sigma^2(\lambda+\tilde y)\bigg)\beta+O(\beta^0)
\end{align}
One cannot satisfy this equation order-by-order in $\beta$. However, a solution
suggests itself: the expansion is singular when $\tilde y=y$. Making this
identification, we find instead
\begin{align}
  &0=\left(4\sigma^2-\frac1{y^2+2(z-\tilde z)}\right)\beta^2+\left(\frac1y+\frac{2y\tilde z}{(y^2+2(z-\tilde z))^2}-4\sigma^2y\right)\beta+O(\beta^0) \\
  &0=\left(-4\sigma^2+\frac1{y^2+2(z-\tilde z)}\right)\beta^2+\left(-\frac y{y^2+2(z-\tilde z)}-\frac{2y\tilde z}{(y^2+2(z-\tilde z))^2}+4\sigma^2(y+\lambda)\right)\beta+O(\beta^0)
\end{align}
Finally, expanding the equation for $\lambda$ to lowest order, we have
\begin{equation}
  0=\omega+2\sigma^2(\lambda-2y)+O(\beta^{-1})
\end{equation}
Simultaneously solving these five equations stemming from the coefficients of $\beta$ for $y$, $z$, $\tilde z$, and $\lambda$, we have
\begin{align}
  \lambda=-\frac1\sigma\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
  &&
  y=\frac1{2\sigma}\left(\frac{\omega}{2\sigma}-\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}\right)
  \\
  z=\frac1{2\sigma^2}\left(1-\frac{\omega^2}{(2\sigma)^2}\right)
  &&
  \tilde z=\frac1{4\sigma^2}\left(1-\frac{\omega}{2\sigma}\left(\frac\omega{2\sigma}+\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}\right)\right)
\end{align}
Inserting this solution into $\mathcal S$ and taking the limit of $\beta$ to zero, we find
\begin{equation}
  G_\sigma(\omega)=-\frac{\omega}{2\sigma}\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
  +\log\left[
    \frac{\omega}{2\sigma}+\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
  \right]
\end{equation}
This function is plotted in Fig. For $\omega<2\sigma$ $G_\sigma(\omega)$ has an
imaginary part, which makes any additional integral over $\omega$ highly
oscillatory. This indicates that the existence of a marginal minimum for this
parameter value corresponds with a large deviation that grows faster than $N$,
rather like $N^2$, since in this regime the bulk of the average spectrum is
over zero and therefore extensively many eigenvalues have to have large
deviations in order for the smallest eigenvalue to be zero. For
$\omega\geq2\sigma$ this function gives the large deviation function for the
probability of seeing a zero eigenvalue given the shift $\omega$.
$\omega=2\sigma$ is the maximum of the function with a real value, and
corresponds to the intersection of the average spectrum with zero.


\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}
  \Sigma_\textrm{marginal}(E)
  =\operatorname{max}_\omega\big[\Sigma(E,\omega)+G_{\sqrt{f''(1)}}(\omega)\big]
\end{equation}
where the maximum over $\omega$ needs to lie at a real value.

\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}

\begin{equation}
  \mathcal S(C,R,D,W,\hat\beta,\omega)
  =\frac12\frac1n
    \sum_{ab}\left(
      \hat\beta^2f(C_{ab})+(2\hat\beta R_{ab}-D_{ab})f'(C_{ab})+(R_{ab}^2-W_{ab}^2)f''(C_{ab})
    \right)
\end{equation}

\begin{equation}
  \mathcal S(C^{11},R^{11},D^{11},W^{11},\hat\beta)+\mathcal S(C^{22},R^{22},D^{22},W^{22},\hat\beta)
  -\epsilon(r_{12}+r_{21})-\omega_1(r^{11}_d-w^{11}_d)-\omega_2(r^{22}_d-w^{22}_d)+\hat\beta E
  +\frac12\log\det\begin{bmatrix}C^{11}&iR^{11}\\iR^{11}&D^{11}\end{bmatrix}
  +\frac12\log\det\left(
  \begin{bmatrix}C^{22}-q_{12}^2C^{11}&iR^{22}\\iR^{22}&D^{22}\end{bmatrix}
  \right)
  -\log\det(W^{11}W^{22}+W^{12}W^{21})
\end{equation}

\begin{equation}
  \begin{aligned}
    &\sum_a^n\left[\hat q_a(Q^{11}_{aa}+Q^{22}_{aa}-1)-\beta(\omega_1Q^{11}_{aa}+\omega_2Q^{22}_{aa}+2\epsilon Q^{12}_{aa})\right]
    +\lambda(\omega_1Q^{11}_{11}+\omega_2Q^{22}_{11}+2\epsilon Q^{12}_{11}) \\
    &+\sum_{i=1,2}f_i''(1)\left[\beta^2\sum_{ab}^n(Q^{ii}_{ab})^2-2\beta\lambda\sum_a^n(Q^{ii}_{1a})^2+\lambda^2(Q^{ii}_{11})^2\right]
    +\frac12\log\det\begin{bmatrix}
      Q^{11}&Q^{12}\\
      Q^{12}&Q^{22}
    \end{bmatrix}
  \end{aligned}
\end{equation}
\begin{equation}
  \log\det\begin{bmatrix}
    Q^{11}&Q^{12}\\
    Q^{12}&Q^{22}
  \end{bmatrix}
  +\log\det(Q^{11}Q^{22}-Q^{12}Q^{12})
\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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