From 1b2ae5156642cdbfa47958c9d1edbfeab5b4b32d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 30 Jun 2022 18:05:01 +0200
Subject: Lots of interpretation work.

---
 frsb_kac-rice.tex | 62 +++++++++++++++++++++++++++++++------------------------
 1 file changed, 35 insertions(+), 27 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 180798a..3c551d7 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -43,9 +43,9 @@ of small temperature for the lowest states, as it should.
 Here we consider, for definiteness, the  mixed $p$-spin model, itself a particular case 
 of the `Toy Model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}
 \begin{equation}
-  H(s)=\sum_p\frac{a_p^{1/2}}{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
+  H(s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
 \end{equation}
-for $\overline{J^2}=p!/2N^{p-1}$. Then
+for $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$. Then
 \begin{equation}
   \overline{H(s_1)H(s_2)}=Nf\left(\frac{s_1\cdot s_2}N\right)
 \end{equation}
@@ -637,50 +637,58 @@ for different energies and typical vs minima.
 
 \section{Interpretation}
 
+Let $\langle A\rangle$ be average over stationary points with given $\epsilon$ and $\mu$, i.e.,
 \begin{equation}
-  H(s)-h^Ts+g\xi^Ts
-\end{equation}
-Let $\langle A\mid\epsilon,\mu\rangle$ be average over stationary points with given $\epsilon$ and $\mu$, i.e.,
-\begin{equation}
-  \langle A\mid\epsilon,\mu\rangle
+  \langle A\rangle
+  =\frac1{\mathcal N}\sum_{\sigma}A(s_\sigma)
   =\frac1{\mathcal N}
-  \int d\nu(s\mid\epsilon,\mu)\,A(s)
+  \int d\nu(s)\,A(s)
 \end{equation}
 with
 \begin{equation}
-  d\nu(s\mid\epsilon,\mu)=ds\,\delta(N\epsilon-H(s))\delta(\partial H(s)+\mu s)|\det(\partial\partial H(s)+\mu I)|
+  d\nu(s)=ds\,\delta(N\epsilon-H(s))\delta(\partial H(s)+\mu s)|\det(\partial\partial H(s)+\mu I)|
 \end{equation}
+Then
 \begin{equation}
-  \frac1N\|\langle s\mid\epsilon,\mu\rangle\|^2
-  =\lim_{n\to0}\int\prod_\alpha^nd\nu(s_\alpha\mid\epsilon,\mu)\,\left(\frac1{n(n-1)}\sum_{\alpha\neq\beta}\frac{s_\alpha^Ts_\beta}N\right)
-  =\lim_{n\to0}\frac1{n(n-1)}\left\langle\sum_{a\neq b}^nC_{ab}\right\rangle
-  =\int_0^1 dx\,c(x)
+  \begin{aligned}
+    \overline{\frac1{N^p}\sum_{i_1\cdots i_p}\langle s_{i_1}\cdots s_{i_p}\rangle\langle s_{i_1}\cdots s_{i_p}\rangle}
+    =\lim_{n\to0}\overline{\int\prod_\alpha^nd\nu(s_\alpha)\,\frac1{n(n-1)}\sum_{a\neq b}\left(\frac{s_a^Ts_b}N\right)^p} \\
+    =\lim_{n\to0}\frac1{n(n-1)}\sum_{a\neq b}^nC^p_{ab}
+    =\int_0^1 dx\,c^p(x)
+  \end{aligned}
 \end{equation}
 
 \begin{equation}
-  \frac1N\sum_i\frac{\partial\langle s_i\rangle}{\partial h_i}
-  =\lim_{n\to0}\int\prod_\alpha^nd\nu(s_\alpha)\,\left(\frac1n\sum_{\alpha\beta}-i\frac{\hat s_\alpha^Ts_\beta}N\right)
-  =\lim_{n\to0}\frac1n\left\langle\sum_{\alpha\beta}R_{\alpha\beta}\right\rangle
+  \begin{aligned}
+    \overline{\frac1{N^p}\sum_{i_1\cdots i_p}\frac{\partial\langle s_{i_1}\cdots s_{i_p}\rangle}{\partial J^{(p)}_{i_1\cdots i_p}}}
+    =\lim_{n\to0}\overline{\int\prod_\alpha^nd\nu(s_\alpha)\,\frac1n\sum_{ab}\left[
+      \hat\beta\left(\frac{s_a^Ts_b}N\right)^p+
+      p\left(-i\frac{\hat s_a^Ts_b}N\right)\left(\frac{s_a^Ts_b}N\right)^{p-1}
+    \right]} \\
+    =\lim_{n\to0}\frac1n\sum_{ab}(\hat\beta C_{ab}^p+pR_{ab}C_{ab}^{p-1})
+    =\hat\beta+pr_d-\int_0^1dx\,c^{p-1}(x)(\hat\beta c(x)+pr(x))
+  \end{aligned}
+\end{equation}
+In particular, when the energy is unconstrained ($\hat\beta=0$),
+\begin{equation}
+  \frac1N\sum_i\frac{\partial\langle s_i\rangle}{\partial J_i^{(1)}}
   =r_d-\int_0^1dx\,r(x)
 \end{equation}
+i.e., adding a linear field causes a response in the average saddle location proportional to $r_d$.
 
 \begin{equation}
   \begin{aligned}
-    \lim_{g\to0}\overline{\frac{\partial^2\Sigma}{\partial g^2}}
-    =\frac1N\lim_{g\to0}\lim_{n\to0}\frac1n\overline{\int\prod_\alpha d\nu(s_\alpha)\left(\sum_\alpha i\xi^T\hat s_\alpha\right)^2}
-    =\lim_{n\to0}\frac1n\int\prod_\alpha d\nu(s_\alpha)\left(\sum_{ab}-\frac{\hat s_a^T\hat s_b}N\right) \\
-    =-\lim_{n\to0}\frac1n\left\langle\sum_{ab}D_{ab}\right\rangle
-    =-d_d+\int_0^1dx\,d(x)
+    \frac{\partial\Sigma}{\partial a_p}
+    =\frac14\lim_{n\to0}\frac1n\sum_{ab}^n\left[
+      \hat\beta^2C_{ab}^p+p(2\hat\beta R_{ab}-D_{ab})C_{ab}^{p-1}+p(p-1)R_{ab}^2C_{ab}^{p-2}
+    \right]
   \end{aligned}
 \end{equation}
-
- The
-meaning of $R_{ab}$ is that of a response of replica $a$ to a linear field in
-replica $b$:
+In particular, when the energy is unconstrained ($\hat\beta=0$),
 \begin{equation}
-    R_{ab} = \frac 1 N \sum_i \overline{\frac{\delta s_i^a}{\delta h_i^b}}
+  \frac{\partial\Sigma}{\partial a_1}=-\frac14\lim_{n\to0}\frac1n\sum_{ab}D_{ab}=-\frac14d_d-\frac14\int_0^1dx\,d(x)
 \end{equation}
-The $D$ may similarly be seen as the variation of the complexity with respect to a random field.
+i.e., adding a linear field decreases the complexity of solutions by an amount proportional to $d_d$.
 
 
 \section{Ultrametricity rediscovered}
-- 
cgit v1.2.3-70-g09d2

