From f40eeebb13e3cd298f9a0b52f04ec88efd214cfd Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 7 Oct 2023 22:31:08 +0200
Subject: Some work.

---
 marginal.tex | 97 +++++++++++++++++++++++++++++++++++++++++++++++++++++++-----
 1 file changed, 89 insertions(+), 8 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 5e67500..e9dc5a8 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -40,10 +40,10 @@
 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)
+  =\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'}}
+  \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 $x_\mathrm{min}(A)$
@@ -57,6 +57,81 @@ 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
+      +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_1^2)
+\end{equation}
+The integral is then given by its value at the stationary point of this
+expression with respect to its three arguments. Making the extremization and taking the limit of $\beta$ to infinity, 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}
+where the branch of the square roots is the same as the sign of $\omega-2\sigma$.
+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)
@@ -144,13 +219,10 @@ describes overlaps between eigenvectors at different stationary points and shoul
 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}
+  \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}
 
@@ -187,6 +259,15 @@ $\Omega=S^{N-1}\times S^{N-1}$
   =Nf_s\left(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\right)
 \end{equation}
 
+\begin{equation}
+  \mathcal S_1(C^{11},R^{11},D^{11},\hat\beta,\omega_1)+\mathcal S_2(C^{22},R^{22},D^{22},\hat\beta,\omega_2)
+  -\epsilon r_{12}-\epsilon r_{21}-\omega_1r^{11}_d-\omega_2r^{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)
+\end{equation}
+
 \section{Multi-species spherical model}
 
 We consider models whose configuration space consists of the product of $r$
-- 
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