From 5a63feda3ce29737c288652f8979cdf3ecb7be39 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 5 Jun 2022 08:39:40 +0200
Subject: Added main result section near the front, and tweaked the conventions
 so that everything is positive in the region of interest.

---
 frsb_kac-rice.tex | 111 ++++++++++++++++++++++++++++++++++++++++++------------
 1 file changed, 86 insertions(+), 25 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index a8f7188..3c6276d 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -88,8 +88,77 @@ To constrain the model to the sphere, we use a Lagrange multiplier $\mu$, with t
   H(s)+\frac\mu2(N-s\cdot s)
 \end{equation}
 
+At any critical point, the hessian is
+\begin{equation}
+  \operatorname{Hess}H=\partial\partial H-\mu I
+\end{equation}
+$\partial\partial H$ is a GOE matrix with variance
+\begin{equation}
+  \overline{(\partial_i\partial_jH)^2}=\frac1Nf''(1)\delta_{ij}
+\end{equation}
+and therefore its spectrum is given by the Wigner semicircle with radius $\sqrt{4f''(1)}$, or
+\begin{equation}
+  \rho(\lambda)=\frac1{\pi\sqrt{f''(1)}}\sqrt{\lambda^2-4f''(1)}
+\end{equation}
+and the spectrum of $\operatorname{Hess}H$ is this shifted by $\mu$, or $\rho(\lambda-\mu)$.
+
+The parameter $\mu$ fixes the spectrum of the hessian. By manipulating it, one
+can decide to find the complexity of saddles of a certain macroscopic index, or
+of minima with a certain harmonic stiffness. When $\mu$ is taken to be within
+the range $\pm\sqrt{4f''(1)}=\pm\mu_m$, the critical points are constrained to have
+index $\frac12N(1-\mu/\mu_m)$. When $\mu>\mu_m$, the critical
+points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally,
+when $\mu=\mu_m$, the critical points are marginal minima.
 
 
+\section{Main result}
+
+\begin{equation}
+  \begin{aligned}
+    \overline{\Sigma(\epsilon,\mu)}
+    =\mathcal D(\mu)
+    +\operatorname*{extremum}_{\substack{R_d,D_d,\hat\epsilon\in\mathbb R\\\chi\in\Lambda}}
+    \left\{
+      \hat\epsilon\epsilon-\mu R_d
+      +\frac12(2\hat\epsilon R_d-D_d)f'(1)+\frac12R_d^2f''(1)
+      +\log R_d \right.\\\left.
+      +\frac12\int_0^1dq\,\left(
+        \hat\epsilon^2f''(q)\chi(q)+\frac1{\chi(q)+R_d^2/D_d}
+      \right)
+    \right\}
+  \end{aligned}
+\end{equation}
+where
+\begin{equation}
+  \mathcal D(\mu)
+  =\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{equation}
+and $\Lambda$ is the space of functions $\chi:[0,1]\to[0,1]$ which are
+monotonically decreasing, convex, and have $\chi(1)=0$ and $\chi'(1)=-1$.
+If there is more than one extremum of this function, choose the one with the
+smallest value of $\Sigma$. The sign of the root inside $\mathcal D(\mu)$ is
+negative for $\mu>0$ and positive for $\mu<0$.
+
+The $k$-RSB ansatz is equivalent to piecewise linear $\chi$ with $k+1$
+pieces, with replica symmetric or 0-RSB giving $\chi(q)=1-q$. Our other major
+result is that, if the equilibrium state in the vicinity of zero temperature is
+given by a $k$-RSB ansatz, then the complexity is given by a $(k-1)$-RSB
+ansatz. Moreover, there is an exact correspondence between the parameters of
+the equilibrium saddle point in the limit of zero temperature and those of the
+complexity saddle saddle at the ground state. If the equilibrium is given by $x_1,\ldots,x_k$ and $q_1,\ldots,q_k$, then the parameters $\tilde x_1,\ldots,\tilde x_{k-1}$ and $\tilde q_1,\ldots,\tilde q_{k-1}$ for the complexity in the ground state are
+\begin{align}
+  \tilde x_i=\frac1{\hat\epsilon}\lim_{\beta\to\infty}\beta x_i
+  &&
+  \tilde q_i=\lim_{\beta\to\infty}q_i
+  &&
+  R_d=\lim_{\beta\to\infty}\beta(1-q_k)
+  &&
+  D_d=R_d\hat\epsilon
+\end{align}
+
 \section{Equilibrium}
 
 Here we review the equilibrium solution. \cite{Crisanti_1992_The, Crisanti_1993_The, Crisanti_2004_Spherical, Crisanti_2006_Spherical}
@@ -209,7 +278,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$.
 
 \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)|
+  =\int ds\,\delta(H(s)-N\epsilon)\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)
@@ -224,7 +293,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$.
   \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)|
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\prod_a^n ds_a\,\delta(H(s_a)-N\epsilon)\delta(\partial H(s_a)-\mu s_a)|\det(\partial\partial H(s_a)-\mu I)|
   \end{aligned}
 \end{equation}
 
@@ -233,7 +302,7 @@ 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)}
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\left(\prod_a^nds_a\right)\,\overline{\prod_a^n \delta(H(s_a)-N\epsilon)\delta(\partial H(s_a)-\mu s_a)}
     \times
     \overline{\prod_a^n |\det(\partial\partial H(s_a)-\mu I)|}
   \end{aligned}
@@ -251,46 +320,38 @@ for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$.
 
 all saddles versus only minima
 
-The parameter $\mu$ fixes the spectrum of the hessian. By manipulating it, one
-can decide to find the complexity of saddles of a certain macroscopic index, or
-of minima with a certain harmonic stiffness. When $\mu$ is taken to be within
-the range $\pm2\sqrt{f''(1)}=\pm\mu_m$, the critical points are constrained to have
-index $\frac12N(1-\mu/\mu_m)$. When $\mu<-\mu_m$, the critical
-points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally,
-when $\mu=\mu_m$, the critical points are marginal minima.
-
 \begin{equation}
-  \prod_a^n\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)
+  \prod_a^n\delta(H(s_a)-N\epsilon)\delta(\partial H(s_a)-\mu s_a)
   =\int \frac{\hat\epsilon}{2\pi}\prod_a^n\frac{d\hat s_a}{2\pi}
-    e^{\hat\epsilon(N\epsilon-H(s_a))+\hat s_a\cdot(\partial H(s_a)-\mu s_a)}
+    e^{\hat\epsilon(H(s_a)-N\epsilon)+i\hat s_a\cdot(\partial H(s_a)-\mu s_a)}
 \end{equation}
 
 \begin{equation}
   \begin{aligned}
     \overline{
       \exp\left\{
-        \sum_a^n(\hat s_a\cdot\partial_a-\hat\epsilon)H(s_a)
+        \sum_a^n(i\hat s_a\cdot\partial_a+\hat\epsilon)H(s_a)
       \right\}
     }
     &=\exp\left\{
         \frac12\sum_{ab}^n
-        (\hat s_a\cdot\partial_a-\hat\epsilon)
-        (\hat s_b\cdot\partial_b-\hat\epsilon)
+        (i\hat s_a\cdot\partial_a+\hat\epsilon)
+        (i\hat s_b\cdot\partial_b+\hat\epsilon)
         \overline{H(s_a)H(s_b)}
       \right\} \\
     &=\exp\left\{
         \frac N2\sum_{ab}^n
-        (\hat s_a\cdot\partial_a-\hat\epsilon)
-        (\hat s_b\cdot\partial_b-\hat\epsilon)
+        (i\hat s_a\cdot\partial_a+\hat\epsilon)
+        (i\hat s_b\cdot\partial_b+\hat\epsilon)
         f\left(\frac{s_a\cdot s_b}N\right)
       \right\} \\
     &\hspace{-13em}\exp\left\{
         \frac N2\sum_{ab}^n
         \left[
           \hat\epsilon^2f\left(\frac{s_a\cdot s_b}N\right)
-          -2\hat\epsilon\frac{\hat s_a\cdot s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
-          +\frac{\hat s_a\cdot \hat s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
-          +\left(\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right)
+          +2i\hat\epsilon\frac{\hat s_a\cdot s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
+          -\frac{\hat s_a\cdot \hat s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
+          -\left(\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right)
         \right]
       \right\}
   \end{aligned}
@@ -299,7 +360,7 @@ when $\mu=\mu_m$, the critical points are marginal minima.
 The parameters:
 \begin{align}
   Q_{ab}=\frac1Ns_a\cdot s_b &&
-  R_{ab}=\frac1N\hat s_a\cdot s_b &&
+  R_{ab}=i\frac1N\hat s_a\cdot s_b &&
   D_{ab}=\frac1N\hat s_a\cdot\hat s_b
 \end{align}
 
@@ -308,10 +369,10 @@ The parameters:
   =\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})
+      \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}
+  +\frac12\log\det\begin{bmatrix}Q&-iR\\-iR&D\end{bmatrix}
   \right)
 \end{equation}
 
-- 
cgit v1.2.3-70-g09d2

