From 2f17fb311918383785ab053eb73e83be803ec8ae Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 4 Sep 2024 02:32:57 +0200
Subject: More work.

---
 topology.tex | 79 +++++++++++++++++++++++++++++++++++-------------------------
 1 file changed, 46 insertions(+), 33 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 884796c..9d0428a 100644
--- a/topology.tex
+++ b/topology.tex
@@ -277,7 +277,7 @@ form
   =\left(\frac N{2\pi}\right)^2\int dR\,dD\,dm\,d\hat m\,g(R,D,m,\hat m)\,e^{N\mathcal S_\Omega(R,D,m,\hat m)}
 \end{equation}
 where $g$ is a prefactor of $o(N^0)$, and $\mathcal S_\Omega$ is an effective action defined by
-\begin{equation}
+\begin{equation} \label{eq:euler.action}
   \begin{aligned}
     \mathcal S_\Omega(R,D,m,\hat m\mid\alpha,V_0)
     &=\hat m-\frac\alpha2\left[
@@ -428,12 +428,12 @@ around the determinant of the Hessian.
 
 Understanding the number of stationary points as a function of latitude $m$
 will clarify the meaning of our effective action for the average Euler
-characteristic. This is because the average number of stationary points is a
+characteristic in the range of overlaps $m$ where it takes a complex value. This is because the average number of stationary points is a
 nonnegative number. If the region of complex $\mathcal S_\Omega$ has a
 well-defined number of stationary points, it indicates that we are looking at a
 situation with a negative average Euler characteristic. On the other hand, if
 the average number of stationary points yields a complex value at some latitude
-$m$, it must be because it is either to large or small in $N$ to be captured by
+$m$, it must be because it is either too large or small in $N$ to be captured by
 the calculation, e.g., that it behaves like $e^{N^2\Sigma}$ or
 $e^{-N^2\Sigma}$. The following calculation indicates this second situation:
 the region of complex action is due to a lack of stationary points to
@@ -620,7 +620,6 @@ realizations of the functions $V_k$ the set $\Omega$ is empty.
 
 \section{Implications for dynamic thresholds in the spherical spin glasses}
 
-\cite{Folena_2020_Rethinking, Folena_2021_Gradient}
 
 As indicated earlier, for $M=1$ the solution manifold corresponds to the energy
 level set of a spherical spin glass with energy density $E=\sqrt NV_0$. All the
@@ -644,7 +643,20 @@ However, for general mixed models the threshold energy is
 \begin{equation}
   E_\mathrm{th}=\pm\frac{f(1)[f''(1)-f'(1)]+f'(1)^2}{f'(1)\sqrt{f''(1)}}
 \end{equation}
-which generally satisfies $|E_\text{shatter}|\leq|E_\text{th}|$.
+which satisfies $|E_\text{shatter}|\leq|E_\text{th}|$. Therefore, as one
+descends in energy in generic models one will meet the shattering energy before
+the threshold energy. This is perhaps unexpected, since one wight imagine that
+where level sets of the energy break into many pieces would coincide with the
+largest concentration of shallow minima in the landscape. We see here that this isn't the case.
+
+This fact mirrors another another that was made clear
+recently: when gradient decent dynamics are run on these models, they will
+asymptotically reach an energy above the threshold energy
+\cite{Folena_2020_Rethinking, Folena_2021_Gradient, Folena_2023_On}. The old
+belief that the threshold energy qualitatively coincides with a kind of
+shattering is the origin of the expectation that the dynamic limit should be
+the threshold energy. Given that our work shows the actual shattering energy is
+different, here we compare it with data on the asymptotic limits of dynamics to see if they coincide.
 
 \begin{figure}
   \includegraphics{figs/dynamics_2.pdf}
@@ -671,7 +683,9 @@ which generally satisfies $|E_\text{shatter}|\leq|E_\text{th}|$.
 \cite{Franz_2016_The, Franz_2017_Universality, Franz_2019_Critical, Annesi_2023_Star-shaped, Baldassi_2023_Typical}
 
 \paragraph{Acknowledgements}
-The authors thank Pierfrancesco Urbani for helpful conversations on these topics.
+The authors thank Pierfrancesco Urbani for helpful conversations on these
+topics, and Giampaolo Folena for supplying his \textsc{dmft} data for the
+spherical spin glasses.
 
 \paragraph{Funding information}
 JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
@@ -831,29 +845,28 @@ We can treat the integral over $\sigma_0$ immediately. It gives
 \end{equation}
 This therefore sets $C=1$ and $G=-R$ in the remainder of the integrand, as well
 as removing all dependence on $\bar H$ and $H$. With these solutions inserted, the remaining terms in the exponential give
-\begin{equation} \label{eq:sdet.q}
-  \begin{aligned}
-    \operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
-    &=1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
-    -\frac6{R^4}\bar H_0H_0\bar{\hat H}\hat H
-    \\
-    &\qquad+\frac2{R^3}\left[
-      (mR-\hat m)(\bar{\hat H}H_0+\bar H_0\hat H)
-      -(D+R^2)\bar H_0H_0
-      +(1-m^2)\bar{\hat H}\hat H
-    \right]
-  \end{aligned}
-\end{equation}
-\begin{equation} \label{eq:sdet.fq}
-  \operatorname{sdet}f(\mathbb Q)
+\begin{align} \label{eq:sdet.q}
+  &\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
+  =1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
+  -\frac6{R^4}\bar H_0H_0\bar{\hat H}\hat H
+  \\
+  &\hspace{5pc}+\frac2{R^3}\left[
+    (mR-\hat m)(\bar{\hat H}H_0+\bar H_0\hat H)
+    -(D+R^2)\bar H_0H_0
+    +(1-m^2)\bar{\hat H}\hat H
+  \right] \notag
+  \\ \label{eq:sdet.fq}
+  &\operatorname{sdet}f(\mathbb Q)
   =1+\frac{Df(1)}{R^2f'(1)}
   +\frac{2f(1)}{R^3f'(1)}\bar{\hat H}\hat H
-\end{equation}
-\begin{equation} \label{eq:inv.q}
-  \int d1\,d2\,f(\mathbb Q)^{-1}(1,2)
+  \\ \label{eq:inv.q}
+  &\int d1\,d2\,f(\mathbb Q)^{-1}(1,2)
   =\left(f(1)+\frac{R^2f'(1)}{D}\right)^{-1}
   +2\frac{Rf'(1)}{(Df(1)+R^2f'(1))^2}\bar{\hat H}\hat H
-\end{equation}
+\end{align}
+The Grassmann terms in these expressions do not contribute to the effective
+action, but will be important in our derivation of the prefactor for the
+calculation in the connected regime. The substitution of these expressions into \eqref{eq:post.hubbard-strat} without the Grassmann terms yields \eqref{eq:euler.action} from the main text.
 
 \section{Calculation of the prefactor of the average Euler characteristic}
 \label{sec:prefactor}
@@ -864,7 +877,7 @@ superspace measure, super-Gaussian integrals do not produce such factors in our
 derivation. Prefactors to our calculation come from three sources: the
 introduction of $\delta$-functions to define the order parameters, integrals
 over Grassmann order parameters, and from the saddle point approximation to the
-large-$N$ integral.
+large-$N$ integral. In addition, there are important contributions of a sign of the magnetization at the solution that arise from our super-Gaussian integrations.
 
 \subsection{Contribution from the Hubbard--Stratonovich transformations}
 \label{sec:prefactor.hs}
@@ -880,7 +893,7 @@ where three factors of $2\pi$ come from the measures as defined in
 integral yields
 \begin{equation}
   \begin{aligned}
-    1=\int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\,d\tilde{\mathbb M}\,
+    1=\frac12\int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\,d\tilde{\mathbb M}\,
     \exp\Bigg\{
       \frac12\int d1\,d2\,\tilde{\mathbb Q}(1,2)\big(N\mathbb Q(1,2)-\pmb\phi(1)\cdot\pmb\phi(2)\big) \qquad \\
       +\int d1\,i\tilde{\mathbb M}(1)\big(N\mathbb M(1)-\mathbf x_0\cdot\pmb\phi(1)\big)
@@ -891,10 +904,10 @@ where the supervectors and measures for $\tilde{\mathbb Q}$ and $\tilde{\mathbb
 analogously to those of $\mathbb Q$ and $\mathbb M$. This is now a super-Gaussian integral in $\pmb\phi$, which can be performed to yield
 \begin{equation}
   \begin{aligned}
-  \int d\pmb\phi\,1=\int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\,d\tilde{\mathbb M}\,
+  \int d\pmb\phi\,1=\frac12\int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\,d\tilde{\mathbb M}\,
   \exp\Bigg\{
     \frac N2\int d1\,d2\,\tilde{\mathbb Q}(1,2)\mathbb Q(1,2)
-    +N\int d1\,i\tilde{\mathbb M}(1)\mathbb M(1) \qquad \\
+    +N\int d1\,i\tilde{\mathbb M}(1)\mathbb M(1) \quad \\
     -\frac N2\log\operatorname{sdet}\tilde{\mathbb Q}
     -\frac N2\int d1\,d2\,\tilde{\mathbb M}(1)\tilde{\mathbb Q}^{-1}(1,2)\tilde{\mathbb M}(2)
   \Bigg\}
@@ -904,7 +917,7 @@ We can perform the remaining Gaussian integral in $\tilde{\mathbb M}$ to find
 \begin{equation}
   \begin{aligned}
     \int d\pmb\phi\,1=
-    \int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\,
+    \frac12\int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\,
     (\operatorname{sdet}\tilde{\mathbb Q}^{-1})^{-\frac12}
     \exp\Bigg\{
       -\frac N2\log\operatorname{sdet}\tilde{\mathbb Q}
@@ -927,10 +940,10 @@ $\frac12\log\det(\mathbb Q-\mathbb M\mathbb M^T)$ in the effective action from
 \end{equation}
 The Hubbard--Stratonovich transformation therefore contributes a factor of
 \begin{equation}
-  \operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)^{\frac12}
-  =[(C-m^2)(D+\hat m^2)+(R-m\hat m)^2]^\frac12G^{-1}
+  \frac12\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)^{\frac12}
+  =\frac12[(C-m^2)(D+\hat m^2)+(R-m\hat m)^2]^\frac12G^{-1}
 \end{equation}
-to the prefactor.
+to the prefactor at the largest order in $N$.
 
 \subsection{Sign of the prefactor}
 
-- 
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