From bfd521829607297717254f8597f39cb94d5b3ee6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 7 Sep 2024 14:50:13 +0200
Subject: Lots of work in the appendicies.

---
 topology.bib |  27 +++++++
 topology.tex | 231 +++++++++++++++++++++++++++++++++++++++++++++++------------
 2 files changed, 212 insertions(+), 46 deletions(-)

diff --git a/topology.bib b/topology.bib
index 88f1a7f..6d09b4e 100644
--- a/topology.bib
+++ b/topology.bib
@@ -265,6 +265,19 @@
  keyword = {Algebraic topology}
 }
 
+@article{Kac_1943_On,
+ author = {Kac, M.},
+ title = {On the average number of real roots of a random algebraic equation},
+ journal = {Bulletin of the American Mathematical Society},
+ publisher = {American Mathematical Society},
+ year = {1943},
+ month = {4},
+ number = {4},
+ volume = {49},
+ pages = {314--320},
+ url = {https://projecteuclid.org:443/euclid.bams/1183505112}
+}
+
 @article{Kamali_2023_Dynamical,
  author = {Kamali, Persia Jana and Urbani, Pierfrancesco},
  title = {Dynamical mean field theory for models of confluent tissues and beyond},
@@ -382,6 +395,20 @@
  eprinttype = {arxiv}
 }
 
+@article{Rice_1939_The,
+ author = {Rice, S. O.},
+ title = {The Distribution of the Maxima of a Random Curve},
+ journal = {American Journal of Mathematics},
+ publisher = {JSTOR},
+ year = {1939},
+ month = {4},
+ number = {2},
+ volume = {61},
+ pages = {409},
+ url = {https://doi.org/10.2307%2F2371510},
+ doi = {10.2307/2371510}
+}
+
 @article{Urbani_2023_A,
  author = {Urbani, Pierfrancesco},
  title = {A continuous constraint satisfaction problem for the rigidity transition in confluent tissues},
diff --git a/topology.tex b/topology.tex
index 6570ca4..a401a65 100644
--- a/topology.tex
+++ b/topology.tex
@@ -130,7 +130,7 @@ solution set. The topological properties revealed by this calculation yield
 surprising results for the well-studied spherical spin glasses, where a
 topological transition thought to occur at a threshold energy $E_\text{th}$
 where marginal minima are dominant is shown to occur at a different energy
-$E_\text{shatter}$. We conjecture that this difference resolves an outstanding
+$E_\text{sh}$. We conjecture that this difference resolves an outstanding
 problem in gradient descent dynamics in these systems.
 
 We consider the problem of finding configurations $\mathbf x\in\mathbb R^N$
@@ -186,8 +186,7 @@ intersect, nor are there self-intersections, without extraordinary fine-tuning.}
 We study the topology of the manifold $\Omega$ by two related means: its
 average Euler characteristic, and the average number of stationary points of a
 linear height function restricted to the manifold. These measures tell us
-complementary pieces of information, respectively the alternating sum and
-direct sum of the Betti numbers of $\Omega$. We find that for the varied cases
+complementary pieces of information. We find that for the varied cases
 we study, these two always coincide at the largest exponential order in $N$,
 putting strong constraints on the resulting topology and geometry.
 
@@ -212,9 +211,9 @@ the count of stationary points of $H$ with increasing index, or
 \begin{equation}
   \chi(\Omega)=\sum_{i=0}^N(-1)^i\mathcal N_H(\text{index}=i)
 \end{equation}
-Conveniently, we can express this abstract sum as an integral over the manifold
+Conveniently, we can express this sum as an integral over the manifold
 using a small variation on the Kac--Rice formula for counting stationary
-points. Since the sign of the determinant of the Hessian matrix of $H$ at a
+points \cite{Kac_1943_On, Rice_1939_The}. Since the sign of the determinant of the Hessian matrix of $H$ at a
 stationary point is equal to its index, if we count stationary points including
 the sign of the determinant, we arrive at the Euler characteristic, or
 \begin{equation} \label{eq:kac-rice}
@@ -379,6 +378,7 @@ These transition values of the target $V_0$ correspond with transition values in
   \includegraphics{figs/action_3.pdf}
 
   \caption{
+    \textbf{Effective action for the Euler characteristic.}
     The effective action governing the average Euler characteristic as a function of the overlap
     $m=\frac1N\mathbf x\cdot\mathbf x_0$ with the height direction for two
     different homogeneous polynomial functions and a variety of target values $V_0$. In both
@@ -585,8 +585,7 @@ realizations of the functions $V_k$ the set $\Omega$ is empty.
   \includegraphics{figs/bar.pdf}
 
   \caption{
-    Cartoon of the topology of the solution manifold implied by our
-    calculation. The arrow shows the vector $\mathbf x_0$ defining the height
+    \textbf{Cartoon of the solution manifold.} The arrow shows the vector $\mathbf x_0$ defining the height
     function. For $V_0<V_\text{on}$, the manifold has a single connected
     component. Above the onset with $V_\text{on}<V_0<V_\text{sh}$, the manifold
     has a large connected component around the equator, and many disconnected
@@ -606,9 +605,10 @@ realizations of the functions $V_k$ the set $\Omega$ is empty.
   \includegraphics{figs/phases_3.pdf}
 
   \caption{
+    \textbf{Topological phase diagram.}
     Topological phases of the model for three different homogeneous covariance
     functions. The onset transition $V_\text{onset}$, shattering transition
-    $V_\text{shatter}$, and satisfiability transition $V_\text{\textsc{sat}}$
+    $V_\text{sh}$, and satisfiability transition $V_\text{\textsc{sat}}$
     are indicated when they exist. In the limit of $\alpha\to0$, the behavior
     of level sets of the spherical spin glasses are recovered: the final plot
     shows how in the pure cubic model the ground state energy $E_\text{gs}$ and threshold energy
@@ -627,15 +627,15 @@ level set of a spherical spin glass with energy density $E=\sqrt NV_0$. All the
 results from the previous sections follow, and can be translated to the spin
 glasses by taking the limit $\alpha\to0$ while scaling $V_0=\sqrt\alpha E$. With a little algebra this procedure yields
 \begin{equation}
-  E_\text{onset}=\pm\sqrt{2f(1)}
+  E_\text{on}=\pm\sqrt{2f(1)}
 \end{equation}
 \begin{equation}
-  E_\text{shatter}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)}
+  E_\text{sh}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)}
 \end{equation}
 for the energies at which level sets of the spherical spin glasses have
 disconnected pieces appear, and that at which a large connected component
 vanishes. For the pure $p$-spin spherical spin glasses with $f(q)=\frac12q^p$,
-$E_\text{shatter}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
+$E_\text{sh}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
 models. This is expected, since threshold energy, defined as the place where
 marginal minima are dominant in the landscape, is widely understood as the
 place where level sets are broken into pieces.
@@ -644,7 +644,7 @@ 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 satisfies $|E_\text{shatter}|\leq|E_\text{th}|$. Therefore, as one
+which satisfies $|E_\text{sh}|\leq|E_\text{th}|$. 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
@@ -665,7 +665,7 @@ different, here we compare it with data on the asymptotic limits of dynamics to
   \includegraphics{figs/dynamics_3.pdf}
 
   \caption{
-    Comparison of the shattering energy $E_\text{sh}$ with the asymptotic
+    \textbf{Is the shattering energy a dynamic threshold?} Comparison of the shattering energy $E_\text{sh}$ with the asymptotic
     performance of gradient descent from a random initial condition in $p+s$
     models with $p=2$ and $p=3$ and varying $s$. The values of $\lambda$ depend on $p$ and $s$ and are taken from \cite{Folena_2023_On}. The points show the asymptotic performance
     extrapolated using two different methods and have unknown uncertainty, from \cite{Folena_2023_On}. Also
@@ -674,7 +674,7 @@ different, here we compare it with data on the asymptotic limits of dynamics to
     that is dashed on the left plot indicates the continuation of the annealed
     result, whereas the solid portion gives the value calculated with a
     {\oldstylenums 1}\textsc{frsb} ansatz.
-  }
+  } \label{fig:ssg}
 \end{figure}
 
 
@@ -727,7 +727,7 @@ The Euler characteristic can be expressed using these supervectors as
 where $d1=d\bar\theta_1\,d\theta_1$ is the integral over the Grassmann
 indices. Since this is an exponential integrand linear in the Gaussian
 functions $V_k$, we can take their average to find
-\begin{equation}
+\begin{equation} \label{eq:χ.post-average}
   \begin{aligned}
     \overline{\chi(\Omega)}
     =\int d\pmb\phi\,d\pmb\sigma\,\exp\Bigg\{
@@ -939,6 +939,7 @@ $\frac12\log\det(\mathbb Q-\mathbb M\mathbb M^T)$ in the effective action from
   =\left(\operatorname{sdet}_{\{1,2\},\{3,4\}}\tilde{\mathbb Q}^{-1}(3,1)\tilde{\mathbb Q}^{-1}(2,4)\right)^{-\frac12}
   =1
 \end{equation}
+where the final superdeterminant is identically 1 for any superoperator $\tilde{\mathbb Q}$, not just its saddle-point value.
 The Hubbard--Stratonovich transformation therefore contributes a factor of
 \begin{equation}
   \frac12\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)^{\frac12}
@@ -950,7 +951,7 @@ to the prefactor at the largest order in $N$.
 
 The superspace notation papers over some analytic differences between branches
 of the logarithm that are not important for determining the saddle point but
-are important to getting correctly the sign of the prefactor. For instance, consider the superdeterminant of $\mathbb Q$ from \eqref{eq:ops} (dropping the fermionic order parameters for a moment for brevity),
+are important to getting correctly the sign of the prefactor. For instance, consider the superdeterminant of $\mathbb Q$ from \eqref{eq:ops.q} (dropping the fermionic order parameters for a moment for brevity),
 \begin{equation}
   \operatorname{sdet}\mathbb Q=\frac{CD+R^2}{G^2}
 \end{equation}
@@ -1280,44 +1281,182 @@ action as a function of $m$ is also exactly the same. We find two other
 solutions that differ from this one, but with formulae too complex to share in
 the text. One alternate solution has $A_-=0$ and one has $A_-\neq0$.
 
+\begin{figure}
+  \includegraphics{figs/alt_sols_1.pdf}
+  \hfill
+  \includegraphics{figs/alt_sols_2.pdf}
+
+  \caption{
+    \textbf{Choosing the correct saddle-point for the complexity.} In the
+    calculation of the complexity of stationary points in the height function,
+    there are three plausible solutions in some regimes. \textbf{Left:} Plot of
+    the three solutions for $M=1$, $f(q)=\frac12q^3$, and
+    $E=E_\text{on}=3^{-1/2}$. The inset shows detail of the region where these
+    solutions are nonnegative. The solid and dashed lines show the values of
+    $m$ where the $A_-=0$ and $A_-\neq0$ alternative solutions are maximized,
+    respectively. The value $m^*$ maximizing the action corresponding also to
+    $\mathcal S_\chi$ is also marked. \textbf{Right:} Numeric measurement of the
+    average value of $m$ for stationary points found using Newton's method as a
+    function of model size $N$. The thick solid line is a power-law fit to
+    $m-m^*$, while the thin solid and dashed lines show the same values of $m$
+    as in the lefthand plot.
+  } \label{fig:alt.sols}
+\end{figure}
+
+For both alternative solutions, there is a regime of small $V_0$ or $E$ for
+which they are unphysical. In this case the action is everywhere negative
+despite our knowledge from the calculation of the Euler characteristic that the
+manifold exists. However, there is a second regime where they cross the action
+$\mathcal S_\chi$ and become positive, both reaching values of $m$ that are
+less than $m_\text{min}$. An example of this regime for the case with
+$f(q)=\frac12q^3$ and $M=1$ (the spherical spin glass) is shown in the left
+panel of Fig.~\ref{fig:alt.sols}. In order to investigate which solution is
+valid in this regime, we turn to comparison with numeric experiments. We create
+samples of this model at the specified parameters, choose random axis $\mathbf
+x_0$, and use Newton's method with mild damping to find stationary points of
+the Lagrangian starting from random initial conditions. When we find a
+stationary point, we record the value of the overlap $m=\frac1N\mathbf
+x\cdot\mathbf x_0$ between it and the height axis. The right panel of
+Fig.~\ref{fig:alt.sols} shows that the average value of $m$ attained by this
+method as a function of $N$ asymptotically approaches $m^*$, the value implied
+by the Euler characteristic action $\mathcal S_\chi$, and not either of the
+values implied by the alternative solutions. This indicates that $\mathcal
+S_\mathcal N(m)=\mathcal S_\chi(m)$ in all regimes.
 
 \section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models}
 \label{sec:1frsb}
 
-\cite{Kent-Dobias_2023_How}
-\[
-  \chi_0(q)=\frac1{\hat\omega_1}f''(q)^{-1/2}-\frac{r_d^2}{d_d}
-\]
-
-\[
+Here we share how the quenched shattering energy is calculated under a
+{\oldstylenums1}\textsc{frsb} ansatz. To best make contact with prior work on
+the spherical spin glasses, we start with \eqref{eq:χ.post-average}. The
+formula in a quenched calculation is almost the same as that for the annealed,
+but the order parameters $C$, $R$, $D$, and $G$ must be understood as $n\times
+n$ matrices rather than scalars. In principle $m$, $\hat m$, $\omega_0$, $\hat\omega_0$, $\omega_1$, and $\hat\omega_1$ should be considered $n$-dimensional vectors, but since in our ansatz replica vectors are constant we can take them to be constant from the start. We have
+\begin{equation}
+  \begin{aligned}
+    \overline{\log\chi(\Omega)}
+    =\lim_{n\to0}\frac\partial{\partial n}\int dC\,dR\,dD\,dG\,dm\,d\hat m\,d\omega_0\,d\hat\omega_0\,d\omega_1\,d\hat\omega_1\,
+    \\
+    \exp N\Bigg\{
+      n\hat m+\frac i2\hat\omega_0\operatorname{Tr}(C-I)-\omega_0\operatorname{Tr}(G+R)
+      -in\hat\omega_1E_0
+      +\frac12\log\det\begin{bmatrix}
+        C-m^2 & i(R-m\hat m) \\ i(R-m\hat m) & D-\hat m^2
+      \end{bmatrix}
+      \\
+      -\frac12\log G^2-\frac12\sum_{ab}^n\left[
+        \hat\omega_1^2f(C_{ab})
+        +(2i\omega_1\hat\omega_1R_{ab}+\omega_1^2D_{ab})f'(C_{ab})
+        +\omega_1^2(G_{ab}^2-R_{ab}^2)f''(C_{ab})
+      \right]
+    \Bigg\}
+  \end{aligned}
+\end{equation}
+which is completely general for the spherical spin glasses with $M=1$. We now
+make a series of simplifications. Ward identities associated with the BRST
+symmetry possessed by the original action indicate that
+\begin{align}
+  \omega_1D=-i\hat\omega_1R
+  &&
+  G=-R
+  &&
+  \hat m=0
+\end{align}
+Moreover, this problem with $m=0$ has a close resemblance to the complexity of
+the spherical spin glasses. In both, at the supersymmetric saddle point the
+matrix $R$ is diagonal with $R=r_dI$ \cite{Kent-Dobias_2023_How}.
+To investigate the shattering energy, we can restrict to solutions with $m=0$
+and look for the place where such solutions vanish. Inserting these simplifications, we have
+\begin{equation}
+  \begin{aligned}
+    \overline{\log\chi(\Omega)}
+    \propto\lim_{n\to0}\frac\partial{\partial n}\int dC\,dR\,d\hat\omega_0\,d\omega_1\,d\hat\omega_1\,
+    \exp N\Bigg\{
+      \frac i2\hat\omega_0\operatorname{Tr}(C-I)
+      -in\hat\omega_1E
+      \\
+      -i\frac12n\omega_1\hat\omega_1r_df'(1)
+      -\frac12\sum_{ab}^n
+        \hat\omega_1^2f(C_{ab})
+      +\frac12\log\det
+      \left(\frac{-i\hat\omega_1}{\omega_1r_d}C+I\right)
+    \Bigg\}
+  \end{aligned}
+\end{equation}
+If we redefine $\hat\beta=-i\hat\omega_1$ and $\tilde r_d=\omega_1 r_d$, we find
+\begin{equation}
+  \begin{aligned}
+    \overline{\log\chi(\Omega)}
+    \propto\lim_{n\to0}\frac\partial{\partial n}\int dC\,d\hat\beta\,d\tilde r_d\,\hat\omega_0\,
+    \exp N\Bigg\{
+      \frac i2\hat\omega_0\operatorname{Tr}(C-I)
+      +n\hat\beta E
+      \\
+      +n\frac12\hat\beta\tilde r_df'(1)
+      +\frac12\sum_{ab}^n
+        \hat\beta^2f(C_{ab})
+      +\frac12\log\det
+      \left(\frac{\hat\beta}{\tilde r_d}C+I\right)
+    \Bigg\}
+  \end{aligned}
+\end{equation}
+which is exactly the effective action for the supersymmetric complexity in the
+spherical spin glasses when in the regime where minima dominate
+\cite{Kent-Dobias_2023_How}. As the effective action for the Euler characteristic, this expression is valid whether minima dominate or not. Following the same steps as in
+\cite{Kent-Dobias_2023_How}, we can write the continuum version of this action
+for arbitrary \textsc{rsb} structure as
+\begin{equation} \label{eq:cont.action}
+  \overline{\log\chi(\Omega)}=\hat\beta E+\frac12\hat\beta\tilde r_df'(1)
+  +\frac12\int_0^1dq\,\left[
+    \hat\beta^2f''(q)\chi(q)+\frac1{\chi(q)+\tilde r_d\hat\beta^{-1}}
+  \right]
+\end{equation}
+where $\chi(q)=\int_1^qdq'\int_0^{q'}dq''P(q'')$ and $P(q)$ is the distribution of
+off-diagonal elements of the matrix $C$. This action must be extremized over
+the function $\chi$ and the variables $\hat\beta$ and $\tilde r_d$, under the
+constraint that $\chi(q)$ is continuous, that it has $\chi'(1)=-1$, and
+$\chi(1)=0$, necessary for $P$ to be a well-defined probability distribution.
+
+Now the specific form of replica symmetry breaking we expect to see is
+important. We want to study the mixed $2+s$ models in the regime where they may
+have 1-full \textsc{rsb} in equilibrium. For the Euler characteristic like the
+complexity, this will correspond to full \textsc{rsb}, in an analogous way to
+{\oldstylenums1}\textsc{rsb} equilibria give a \textsc{rs} complexity. Such order is characterized by a piecewise smooth $\chi$ of the form
+\begin{equation}
   \chi(q)=\begin{cases}
     \chi_0(q) & q < q_0 \\
-    1-(1-m)q_1-mq & q_0 < q < q_1 \\
-    1-q & q > q_1
+    1-q & q \geq q_0
   \end{cases}
-\]
-\[
-  0=\hat\omega_1r_d-\omega_1d_d
-  \qquad
-  \omega_1=\hat\omega_1\frac{r_d}{d_d}
-\]
-\[
-  \log\chi
-  =-\hat\omega_1 E
-  +\frac12\hat\omega_1^2r_d^2/d_df'(1)
-  +\frac12\int_0^1dq\,\left[
-    \hat\omega_1^2f''(q)\chi(q)
-    +\frac1{\chi(q)+r_d^2/d_d}
-  \right]
-  -\frac12\log r_d^2
-\]
-\[
-  0=-\frac{\hat\omega_1^2f'(1)}{d_d}+\int_0^1dq\,\frac1{(r_d^2/d_d+\chi(q))^2}
-\]
-\[
-  d_d=-\frac{1+r_d}{\int dq\,\chi(q)}r_d
-\]
+\end{equation}
+where $\chi_0$ is
+\begin{equation}
+  \chi_0(q)=\frac1{\hat\beta}[f''(q)^{-1/2}-\tilde r_d]
+\end{equation}
+the function implied by extremizing \eqref{eq:cont.action} over $\chi$. The
+variable $q_0$ must be chosen so that $\chi$ is continuous. The key difference
+between \textsc{frsb} and {\oldstylenums1}\textsc{frsb} in this setting is that
+in the former case the ground state has $q_0=1$, while in the latter the ground
+state has $q_0<1$.
 
+We use this action to find the shattering energy in the following way. First, setting $\overline{\log\chi(\Omega)}=0$ and solving for $E$ yields a formula for the ground state energy
+\begin{equation}
+  E_\text{gs}=-\frac1{\hat\beta}\left\{
+    \frac12\hat\beta\tilde r_df'(1)
+    +\frac12\int_0^1dq\,\left[
+      \hat\beta^2f''(q)\chi(q)+\frac1{\chi(q)+\tilde r_d\hat\beta^{-1}}
+    \right]
+  \right\}
+\end{equation}
+This expression can be maximized over $\hat\beta$ and $\tilde r_d$ to find the
+correct parameters at the ground state for a particular model. Then, the
+shattering energy is found by slowly lowering $q_0$ and solving the combined
+extremal and continuity problem for $\hat\beta$, $\tilde r_d$, and $E$ until
+$E$ reaches a maximum value and starts to decrease. This maximum is the
+shattering energy, since it is the point where the $m=0$ solution vanishes.
+Starting from this point, we take small steps in $s$ and $\lambda_s$, again
+simultaneously extremizing, ensuring continuity, and maximizing $E$. This draws
+out the shattering energy across the entire range of $s$ plotted in
+Fig.~\ref{fig:ssg}. The transition to the \textsc{rs} solution occurs when the value $q_0$ that maximizes $E$ hits zero.
 
 \bibliography{topology}
 
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