From 891f03d9ec851c5a5ae58ce4aa7dcdd7504f2ff8 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 6 Jun 2024 08:22:06 +0200
Subject: More writing.

---
 marginal.tex | 130 ++++++++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 115 insertions(+), 15 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 5c3e033..7df99d5 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -607,16 +607,57 @@ which gives $\mu_m(E)=2\sqrt{f''(1)}$ independent of $E$, as we presaged above.
 that the marginal complexity in these models is simply the ordinary complexity
 evaluated at a fixed trace of the Hessian.
 
-\subsection{Twin spherical spin glasses}
-
-$\Omega=S^{N-1}\times S^{N-1}$
-\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}
+\subsection{Multispherical spin glasses}
+
+The multispherical models are a simple extension of the spherical ones, where
+the configuration space is taken to be the union of more than one hypersphere.
+Here we consider the specific case where the configuration space is the union
+of two $(N-1)$-spheres, with $\Omega=S^{N-1}\times S^{N-1}$, and where the
+energy is given by
+\begin{equation}
+  H(\mathbf x)=H_1(\mathbf x^{(1)})+H_2(\mathbf x^{(2)})+\epsilon\mathbf x^{(1)}\cdot\mathbf x^{(2)}
+\end{equation}
+for $\mathbf x=[\mathbf x^{(1)},\mathbf x^{(2)}]$ for components $\mathbf
+x^{(1)},\mathbf x^{(2)}\in\mathbb R^N$. Each individual sphere energy $H_s$ is
+taken to be a centered Gaussian random function with a covariance given in the
+usual spherical way by
+\begin{equation}
+  \overline{H_s(\pmb\sigma_1)H_p(\pmb\sigma_2)}
+  =N\delta_{sp}f_s\left(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\right)
+\end{equation}
+with the functions $f_1$ and $f_2$ not necessarily the same. In this problem,
+there is an energetic competition between the independent spin glass energies
+on each sphere and their tendency to align or anti-align through the
+interaction term.
+
+Because the energy is Gaussian, properties of the Hessian are once again
+statistically independent of those of the energy and gradient. However, unlike
+the previous example of the spherical models, the spectrum of the Hessian at
+different points in the configuration space has different shapes. This appears
+in this problem through the presence of a configuration space defined by
+multiple constraints, and therefore multiple Lagrange multipliers are necessary
+to ensure they are all fixed.
+\begin{align}
+  H(\mathbf x)
+  +\frac12\omega^{(1)}\big(\|\mathbf x^{(1)}\|^2-N\big)
+  +\frac12\omega^{(2)}\big(\|\mathbf x^{(2)}\|^2-N\big)
+  \\
+  \nabla H(\mathbf x,\pmb\omega)
+  =\partial H(\mathbf x)+\begin{bmatrix}
+    \omega^{(1)}\mathbf x^{(1)} \\
+    \omega^{(2)}\mathbf x^{(2)}
+  \end{bmatrix}
+  \\
+  \operatorname{Hess}H(\mathbf x,\pmb\omega)
+  =\partial\partial H(\mathbf x)+\begin{bmatrix}
+    \omega^{(1)}I&0 \\
+    0&\omega^{(2)}I
+  \end{bmatrix}
+\end{align}
+Like in the spherical model, fixing the trace of the Hessian to $\mu$ is
+equivalent to a constraint on the Lagrange multipliers. However, in this case
+it corresponds to $\mu=\omega^{(1)}+\omega^{(2)}$, and therefore they are not
+uniquely fixed by the trace.
 
 \begin{widetext}
 \begin{equation}
@@ -769,19 +810,29 @@ The first step to evaluate this expression is to linearize the dependence on the
     \big(V^k(\pmb\phi_{a\alpha}(1,2))-v_{a\alpha}^k(1,2)\big)
   \right]
 \end{equation}
-where we have introduced auxiliary fields $\hat v$.
+where we have introduced auxiliary fields $\hat v$. With this inserted into the
+integral, all other instances of $V$ are replaced by $v$, and the only
+remaining dependence on the disorder is from the term $\hat vV$ arising from
+the Fourier representation of the Dirac $\delta$ function. This term is linear in $V$, and therefore the random functions can be averaged over to produce
 \begin{equation}
-  -\sum_{ab}\sum_{\alpha\gamma}\sum_k\frac12\int d1\,d2\,d3\,d4\,
-  \hat v_{a\alpha}^kf\big(\pmb\phi_{a\alpha}(1,2)^T\pmb\phi_{b\gamma}(3,4)\big)\hat v_{b\gamma}^k
+  \overline{
+    \exp\left[
+      i\sum_{a\alpha k}\int d1\,d2\,\hat v_{a\alpha}^k(1,2)
+      V^k(\pmb\phi_{a\alpha}(1,2))
+    \right]
+  }
+  =
+  -\frac N2\sum_{ab}^n\sum_{\alpha\gamma}^{m_a}\sum_k^{\alpha N}\int d1\,d2\,d3\,d4\,
+  \hat v_{a\alpha}^k(1,2)f\big(\pmb\phi_{a\alpha}(1,2)^T\pmb\phi_{b\gamma}(3,4)\big)\hat v_{b\gamma}^k(3,4)
 \end{equation}
-We're now quadratic in the $v$ and $\hat v$ with the kernel
+The entire integrand is now quadratic in the $v$ and $\hat v$ with the kernel
 \begin{equation}
   \begin{bmatrix}
     B_{a\alpha}(1,2)\delta(1,3)\delta(2,4)\delta_{ab}\delta_{\alpha\gamma} & i\delta(1,3)\,\delta(2,4) \delta_{ab}\delta_{\alpha\gamma}\\
     i\delta(1,3)\,\delta(2,4) \delta_{ab}\delta_{\alpha\gamma}& f\big(\pmb\phi_{a\alpha}(1,2)^T\pmb\phi_{b\gamma}(3,4)\big)
   \end{bmatrix}
 \end{equation}
-Upon integration, this results in a term in the effective action of the form
+The integration over the $v$ and $\hat v$ results in a term in the effective action of the form
 \begin{equation}
   -\frac M2\log\operatorname{sdet}\left(
     \delta(1,3)\,\delta(2,4) \delta_{ab}\delta_{\alpha\gamma}
@@ -836,6 +887,55 @@ typical pairs of stationary points have no overlap. This gives
 \end{align}
 We further take a planted replica symmetric structure for the matrix $Q$,
 identical to that in \eqref{eq:Q.structure}.
+\begin{equation}
+  \begin{aligned}
+    \mathcal S
+    =-\frac\alpha2\log\left[
+      \frac{
+        (f'(1)d-\hat\beta-f''(1)(r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\hat\lambda)+\beta\hat\lambda+\frac12\hat\lambda^2)))(f(1)-f(0))+(1-rf'(1))^2)
+      }{
+        (1+gf'(1))^2
+      }
+    \right] \\
+    +\frac{\alpha f(0)}2\frac{
+      \hat\beta-df'(1)+(r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\lambda)+\lambda\beta+\frac12\lambda^2)f''(1)
+    }{
+      (f'(1)d-\hat\beta- (r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\hat\lambda)+\beta\hat\lambda+\frac12\hat\lambda^2)f''(1))(f(1)-f(0))+(1-rf'(1))^2
+    }
+    \\
+    -\frac\alpha2\log\left(\frac{1-f'(1)(2\beta(1-q_0)+\lambda-(1-2q_0+\tilde q_0^2)\beta(\beta+\lambda)f'(1))}{(1-(1-q_0)\beta f'(1))^2}\right)
+    +\frac12\log\frac{d+r^2}{g^2}\frac{1-2q_0+\tilde q_0^2}{(1-q_0)^2}
+    -\mu(r+g)-\frac12\mu\hat\lambda+\hat\beta E+\hat\lambda\lambda^*
+  \end{aligned}
+\end{equation}
+\begin{equation}
+  \begin{aligned}
+    \mathcal S_\infty
+    =-\frac\alpha2\log\left[
+      \frac{
+        (f'(1)d-\hat\beta-f''(1)(r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\hat\lambda^2)))(f(1)-f(0))+(1-rf'(1))^2)
+      }{
+        (1+gf'(1))^2
+      }
+    \right] \\
+    +\frac{\alpha f(0)}2\frac{
+      \hat\beta-df'(1)+(r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\lambda^2)f''(1)
+    }{
+      (f'(1)d-\hat\beta- (r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\lambda^2)f''(1))(f(1)-f(0))+(1-rf'(1))^2
+    }
+    \\
+    -\frac\alpha2\log\left(
+      \frac{
+        1-(2y_0+\hat\lambda)f'(1)+(y_0^2-\Delta z)f'(1)^2
+      }{(1-y_0f'(1))^2}
+    \right)
+    +\frac12\log\frac{d+r^2}{g^2}\frac{y_0^2-\Delta z}{y_0^2}
+    -\mu(r+g)-\frac12\mu\hat\lambda+\hat\beta E+\hat\lambda\lambda^*
+  \end{aligned}
+\end{equation}
+\begin{equation}
+  \Sigma_{\lambda^*}(E,\mu)=\operatorname{extremum}_{\hat\beta,r,d,g,y_0,\Delta z,\hat\lambda}\mathcal S_\infty
+\end{equation}
 \end{widetext}
 
 \appendix
-- 
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