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(-)
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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