Lots of equation writing setting up Kac-Rice.

1 files changed, 92 insertions, 13 deletions

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index eb940fa..3e06370 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -28,8 +28,6 @@ Although the Bray-Moore computation for the SK model was the first application o
 some replica symmetry breaking scheme, it turned out that the problem has been open 
 ever since.
 
-
-
 to this date the program has been only complete for a subset of models 
 
 here we present what we believe is the general scheme
@@ -86,7 +84,7 @@ Here we consider, for definiteness, the `toy' model introduced by M\'ezard and P
 
 \section{Equilibrium}
 
-Here we review the equilibrium solution.
+Here we review the equilibrium solution. \cite{Crisanti_1992_The, Crisanti_1993_The, Crisanti_2004_Spherical, Crisanti_2006_Spherical}
 
 \begin{equation}
   \beta F=\frac12\lim_{n\to0}\frac1n\left(\beta^2\sum_{ab}f(Q_{ab})+\log\det Q\right)-\frac12\log S_\infty
@@ -199,28 +197,109 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$.
 
 \section{Kac-Rice}
 
+\cite{Auffinger_2012_Random, BenArous_2019_Geometry}
+
+\begin{equation}
+  \mathcal N(\epsilon, \mu)
+  =\int ds\,\delta(N\epsilon-H(s))\delta(\partial H(s)-\mu s)|\det(\partial\partial H(s)-\mu I)|
+\end{equation}
+\begin{equation}
+  \Sigma(\epsilon,\mu)=\frac1N\log\mathcal N(\epsilon, \mu)
+\end{equation}
+
 \subsection{The replicated  problem}
 
-\begin{eqnarray}
-&=& \Pi_a \delta(Eq_a)  \;  \Pi_a \left| \det_a( )\right| \delta(E_a-E(s_a))\nonumber\\
-&\rightarrow& \overline{\Pi_a \delta(Eq_a)}  \; \overline{ \Pi_a \left| \det_a( )\right|\delta(E_a-E(s_a))}\nonumber\\
-\end{eqnarray}
+\cite{Ros_2019_Complex}
+\cite{Folena_2020_Rethinking}
+
+\begin{equation}
+  \begin{aligned}
+    \Sigma(\epsilon, \mu)
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\mathcal N^n(\epsilon) \\
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\prod_a^n ds_a\,\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)|\det(\partial\partial H(s_a)-\mu I)|
+  \end{aligned}
+\end{equation}
 
-the question of independence
+the question of independence \cite{Bray_2007_Statistics}
+
+\begin{equation}
+  \begin{aligned}
+    \overline{\Sigma(\epsilon, \mu)}
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\left(\prod_a^nds_a\right)\,\overline{\prod_a^n \delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)}
+    \times
+    \overline{\prod_a^n |\det(\partial\partial H(s_a)-\mu I)|}
+  \end{aligned}
+\end{equation}
+
+\begin{equation}
+  \begin{aligned}
+    \mathcal D(\mu)
+    &=\frac1N\overline{\log|\det(\partial\partial H(s_a)-\mu I)|}
+    =\int d\lambda\,\rho(\lambda-\mu)\log|\lambda| \\
+    &=\operatorname{Re}\left\{\frac12\left(1+\frac\mu{2f''(1)}\left(\mu\pm\sqrt{\mu^2-4f''(1)}\right)\right)-\log\left(\frac1{2f''(1)}\left(\mu\pm\sqrt{\mu^2-4f''(1)}\right)\right)\right\}
+  \end{aligned}
+\end{equation}
+for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$.
+
+\begin{equation}
+  \begin{aligned}
+    \overline{\Sigma(\epsilon, \mu)}
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
+    e^{nN\mathcal D(\mu)}
+    \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}\overline{
+      \exp\left[
+        \sum_a^n
+        (\hat s_a\partial_a-\hat\epsilon)H(s_a)
+      \right]
+    }
+    \\
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
+    e^{nN\mathcal D(\mu)}
+    \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}
+      \exp\left[
+        \sum_{ab}^n
+        (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)f(s_as_b/N)
+      \right] \\
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
+    e^{nN\mathcal D(\mu)}
+    \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}
+      \exp\left[
+        \sum_{ab}^n
+        (
+        \hat\epsilon^2f(s_as_b/N)-2\hat\epsilon\hat s_as_bf'(s_as_b/N)+\hat s_a\hat s_bf'(s_as_b/N)
+        +(\hat s_as_b)^2f''(s_as_b/N)
+        )
+      \right]
+  \end{aligned}
+\end{equation}
 
 all saddles versus only minima
 
 The parameters:
-\begin{eqnarray}
-Q_{ab}&=&\nonumber\\
-R_{ab}&=&\nonumber\\
-D_{ab}&=&
-\end{eqnarray}
+\begin{equation}
+  \begin{aligned}
+    Q_{ab}=\frac1Ns_a\cdot s_b \\
+    R_{ab}=\frac1N\hat s_a\cdot s_b \\
+    D_{ab}=\frac1N\hat s_a\cdot\hat s_b
+  \end{aligned}
+\end{equation}
 
+\begin{equation}
+  S
+  =\mathcal D(\mu)+\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left(
+    \mu\sum_a^nR_{aa}
+    +\frac12\sum_{ab}\left[
+      \hat\epsilon^2f(Q_{ab})-2\hat\epsilon R_{ab}f'(Q_{ab})
+      +D_{ab}f'(Q_{ab})+R_{ab}^2f''(Q_{ab})
+    \right]
+  +\frac12\log\det\begin{bmatrix}Q&R\\R&D\end{bmatrix}
+  \right)
+\end{equation}
 
 
 
 \section{Replicated action}
+
 \begin{align*}
   \Sigma
   =-\epsilon\hat\epsilon+\lim_{n\to0}\frac1n\left(