From 0f7a0cab236c3bebca37e340ba36b8cd0bb77120 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 1 Aug 2024 18:29:29 +0200
Subject: Lots more writing.

---
 topology.tex | 90 ++++++++++++++++++++++++++++--------------------------------
 1 file changed, 42 insertions(+), 48 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 6e694a1..89692b7 100644
--- a/topology.tex
+++ b/topology.tex
@@ -30,14 +30,11 @@
   average Euler characteristic of this manifold, and find different behaviors
   depending on $\alpha=M/N$. When $\alpha<1$, the average Euler characteristic
   is subexponential in $N$ but positive, indicating the presence of few
-  simply-connected components. When $1\leq\alpha<\alpha_\mathrm a^*$, it is
+  simply-connected components. When $1\leq\alpha<\alpha_\text{\textsc{sat}}$, it is
   exponentially large in $N$, indicating a shattering transition in the space
-  of solutions. Finally, when $\alpha_\mathrm a^*\leq\alpha$, the number of
+  of solutions. Finally, when $\alpha_\text{\textsc{sat}}\leq\alpha$, the number of
   solutions vanish. We further compute the average logarithm of the Euler
-  characteristic, which is representative of typical manifolds. We compare
-  these results with the analogous calculation for the topology of level sets
-  of the spherical spin glasses, whose connected phase has a negative Euler
-  characteristic indicative of many holes.
+  characteristic, which is representative of typical manifolds.
 \end{abstract}
 
 \maketitle
@@ -325,19 +322,30 @@ $\alpha\geq\alpha_\text{\textsc{sat}}$, there are no longer solutions that
 satisfy the constraints. The Euler characteristic is not defined for an empty
 set, and in this regime the calculation yields no solution.
 
+We have made the above discussion assuming that $\alpha_\text{\textsc{sat}}>1$.
+However, this isn't necessary, and it is straightforward to produce covariance
+functions $f$ where $\alpha_\text{\textsc{sat}}<1$. In this case, the picture
+changes somewhat. When $\alpha_\text{\textsc{sat}}<\alpha<1$, the action
+$\mathcal S_\mathrm a$ has a single maximum at $m^*=0$, where it is negative.
+This corresponds to an average Euler characteristic $\overline\chi$ which is
+exponentially small in $N$. Such a situation is consistent with typical
+constraints leading to no solutions and a zero characteristic, but rare and
+atypical configurations having some solutions.
+
 In the regime where $\log\overline\chi$ is positive, it is possible that our
 calculation yields a value which is not characteristic of typical sets of
 constraints. This motivates computing $\overline{\log\chi}$, the average of
 the logarithm, which should produce something characteristic of typical
-samples, the so-called quenched calculation.
-
-\begin{equation}
-  D=\beta R
-  \qquad
-  \beta=-\frac{m+\sum_aR_{1a}}{\sum_aC_{1a}}
-  \qquad
-  \hat m=0
-\end{equation}
+samples, the so-called quenched calculation. In an appendix to this paper we
+sketch the quenched calculation and report its result in the replica symmetric
+approximation. This differs from the annealed calculation above only when
+$f(0)>0$. The replica symmetric calculation produces the same transitions at
+$\alpha=1$ and $\alpha=\alpha_\text{\textsc{sat}}$, but modifies the value
+$m^*$ in the connected phase and predicts
+$\frac1N\overline{\log\chi}<\frac1N\log\overline\chi$ in the shattered phase.
+The fact that $\alpha_\text{\textsc{sat}}=f'(1)/f(1)$ is the same in the annealed and
+replica symmetric calculations suggests that it may perhaps be exact. It is also
+consistent with the full RSB calculation of \cite{Urbani_2023_A}.
 
 
 \begin{acknowledgements}
@@ -347,45 +355,31 @@ samples, the so-called quenched calculation.
 
 \bibliography{topology}
 
-\appendix
-
-\section{Euler characteristic of the spherical spin glasses}
+\paragraph{Quenched average of the Euler characteristic.}
 
-We can compare this calculation with what we expect to find for the manifold
-defined by $V(\mathbf x)=E$ for a single function $V$, with a rescaled covariance $\overline{V(\mathbf x)V(\mathbf x')}=Nf(\mathbf x\cdot\mathbf x'/N)$. This corresponds to the
-energy level set of a spherical spin glass. Now the Lagrangian is
 \begin{equation}
-  L(\mathbf x,\omega_0,\omega_1)=
-  H(\mathbf x)+\frac12\omega_0\big(\|\mathbf x\|^2-N\big)
-  +\omega_1\big(V(\mathbf x)-NE\big)
+  D=\beta R
+  \qquad
+  \hat\beta=-\frac{m+\sum_aR_{1a}}{\sum_aC_{1a}}
+  \qquad
+  \hat m=0
 \end{equation}
-The derivation follows almost identically as before, but we do not integrate
-out $\sigma_1$. We have
+
 \begin{align}
-  \overline{\chi}&=\int d\mathbb Q\,d\mathbb M\,d\sigma_0\,d\sigma_1\,\exp\Bigg\{
-    \frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
-    \notag \\
-    &\quad+N\int d1\,\bigg[
-      \mathbb M(1)+\frac12\sigma_0(1)\big(\mathbb Q(1,1)-1\big) \notag \\
-    &-E\sigma_1(1)
-      +\frac 12\int d2\,\sigma_1(1)\sigma_1(2)f\big(\mathbb Q(1,2)\big)
-    \bigg]
-  \Bigg\}
+  &\mathcal S(m,C,R)
+  =\frac12\log\det\big[I+\hat\beta R^{-1}(C-m^2)\big] \notag \\
+  &\quad-\frac\alpha2\log\det\big[I+\hat\beta\big(R\odot f'(C)\big)^{-1}f(C)\big]
 \end{align}
-The saddle point condition for $\sigma_1$ gives
-\begin{equation}
-  \sigma_1(1)=E\int d2\,f(\mathbb Q)^{-1}(1,2)
-\end{equation}
-which then yields
+
 \begin{align}
-  \overline{\chi}&=\int d\mathbb Q\,d\mathbb M\,d\sigma_0\,\exp\Bigg\{
-    \frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
-    \notag \\
-    &\quad+N\int d1\,\bigg[
-      \mathbb M(1)+\frac12\sigma_0(1)\big(\mathbb Q(1,1)-1\big) \notag \\
-    &-\frac12E^2\int d2\,f(\mathbb Q)^{-1}(1,2)
-    \bigg]
-  \Bigg\}
+  \mathcal S_0(m,r_d)
+  =\frac12\bigg[
+    -m(m+r_d)+\log\left(-\frac m{r_d}\right) \notag \\
+    -\alpha\log\left(
+      \frac{-m\big(f(1)-f(0)\big)+r_d\big(f'(1)-f(1)+f(0)\big)}{r_df'(1)}
+    \right) \notag \\
+    +\frac{\alpha f(0)(m+r_d)}{-m\big(f(1)-f(0)\big)+r_d\big(f'(1)-f(1)+f(0)\big)}
+  \bigg]
 \end{align}
 
 \end{document}
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