From 87510f78bef9224f38f4588eccecd0d0247aa73d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 21 Oct 2022 15:32:48 +0200
Subject: Added a table for interesting energy values.

---
 frsb_kac-rice_letter.tex | 45 +++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 43 insertions(+), 2 deletions(-)

(limited to 'frsb_kac-rice_letter.tex')

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index ebbadf3..1d8785c 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -119,8 +119,10 @@ is given by
 with two additional $\delta$-functions inserted to fix the energy density $E$
 and the stability $\mu$. The complexity is then
 \begin{equation} \label{eq:complexity}
-  \Sigma(E,\mu)=\lim_{N\to\infty}\frac1N\overline{\log\mathcal N(E, \mu})
+  \Sigma(E,\mu)=\lim_{N\to\infty}\frac1N\overline{\log\mathcal N(E, \mu}).
 \end{equation}
+Most of the difficulty of this calculation resides in the logarithm in this
+formula.
 
 The stability $\mu$, sometimes called the radial reaction, determines the depth
 of minima or the index of saddles. At large $N$ the Hessian can be shown to
@@ -178,6 +180,19 @@ supersymmetry in the Kac--Rice formula, studied in the past in the context of
 the TAP free energy. This line of `supersymmetric' solutions terminates at the
 ground state, and describes the most numerous types of stable minima.
 
+Using this solution, one finds a correspondence between properties of the
+overlap matrix at the ground state energy, where the complexity vanishes,
+and the overlap matrix in the equilibrium problem in the limit of zero
+temperature. The saddle point parameters of the two problems are related
+exactly. In the case where the vicinity of the equilibrium ground state is
+described by a $k$RSB solution, the complexity at the ground state is
+$(k-1)$RSB. This can be intuitively understood by considering the difference
+between measuring overlaps between equilibrium \emph{states} and stationary
+\emph{points}. For states, the finest level of the hierarchical description
+gives the typical overlap between two points drawn from the same state, which
+has some distribution about the ground state at nonzero temperature. For
+points, this finest level does not exist.
+
 In general, solving the saddle-point equations for the parameters of the three
 replica matrices is challenging. Unlike the equilibrium case, the solution is
 not extremal, and so minimization methods cannot be used. However, the line of
@@ -225,7 +240,7 @@ find the complexity everywhere. This is how the data in what follows was produce
 For the first example, we study a model whose complexity has the simplest
 replica symmetry breaking scheme, 1RSB. By choosing a covariance $f$ as the sum
 of polynomials with well-separated powers, one develops 2RSB in equilibrium.
-This should correspond to 1RSB in the complexity. For this example, we take
+This should correspond to 1RSB in the complexity. We take
 \begin{equation}
   f(q)=\frac12\left(q^3+\frac1{16}q^{16}\right)
 \end{equation}
@@ -251,6 +266,32 @@ marginal minima are the most common stationary points. Something about the
 topology of the energy function might be relevant to where this algorithmic threshold
 lies. For the $3+16$ model at hand, $E_\mathrm{alg}=-1.275\,140\,128\ldots$.
 
+\begin{table}
+  \begin{tabular}{l|cc}
+    & $3+16$ & $2+4$ \\\hline\hline
+    $\langle E\rangle_\infty$ &---& $-0.531\,25\hphantom{1\,111\dots}$ \\
+    $\hphantom{\langle}E_\mathrm{max}$ &     $-0.886\,029\,051\dots$ & $-1.039\,701\,412\dots$\\
+    $\langle E\rangle_1$ & $-0.906\,391\,055\dots$ & ---\\
+    $\langle E\rangle_2$ & $-1.195\,531\,881\dots$ & ---\\
+    $\hphantom{\langle}E_\mathrm{dom}$ & $-1.273\,886\,852\dots$ & $-1.056\,6\hphantom{11\,111\dots}$\\
+    $\hphantom{\langle}E_\mathrm{alg}$ & $-1.275\,140\,128\dots$ & $-1.059\,384\,319\ldots$\\
+    $\hphantom{\langle}E_\mathrm{th}$ & $-1.287\,575\,114\dots$ & $-1.059\,384\,319\ldots$\\
+    $\hphantom{\langle}E_0$ & $-1.287\,605\,530\ldots$ & $-1.059\,384\,319\ldots$\\\hline
+  \end{tabular}
+  \caption{
+    Landmark energies of the equilibrium and complexity problems for the two
+    models studied. $\langle E\rangle_1$, $\langle E\rangle_2$ and $\langle
+    E\rangle_\infty$ are the average energies in equilibrium at the RS--1RSB,
+    1RSB--2RSB, and RS--FRSB transitions, respectively. $E_\mathrm{max}$ is the
+    highest energy at which any stationary points are described by a RSB
+    complexity. $E_\mathrm{dom}$ is the energy at which dominant stationary
+    points have an RSB complexity. $E_\mathrm{alg}$ is the algorithmic
+    threshold below which smooth algorithms cannot go. $E_\mathrm{th}$ is the
+    traditional threshold energy, defined by the energy at which marginal
+    minima become most common. $E_0$ is the ground state energy.
+  } \label{tab:energies}
+\end{table}
+
 In this model, the RS complexity gives an inconsistent answer for the
 complexity of the ground state, predicting that the complexity of minima
 vanishes at a higher energy than the complexity of saddles, with both at a
-- 
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