Finishing up the letter.

1 files changed, 44 insertions, 21 deletions

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index 63b7811..4fb22e7 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -27,7 +27,7 @@
 
 \begin{abstract}
   Complexity is a measure of the number of stationary points in complex
-  landscapes. We {\color{red} solve the long-standing problem of detremining the...} derive a general solution for the complexity of mean-field
+  landscapes. We derive a general solution for the complexity of mean-field
   complex landscapes. It incorporates Parisi's solution for the ground state,
   as it should. Using this solution, we count the stationary points of two
   models: one with multi-step replica symmetry breaking, and one with full
@@ -134,14 +134,16 @@ stationary points using a $\delta$-function weighted by a Jacobian
 \begin{equation}
   \begin{aligned}
     \mathcal N(E, \mu)
-    &=\int_{S^{N-1}}d\mathbf s\, \delta\big(\nabla H(\mathbf s)\big)\,\big|\det\operatorname{Hess}H(\mathbf s)\big| \\
+    &=\int_{\mathbb R^N}d\boldsymbol\xi\,e^{-\frac12\|\boldsymbol\xi\|^2/\sigma^2}\int_{S^{N-1}}d\mathbf s\, \delta\big(\nabla H(\mathbf s)-\boldsymbol\xi\big)\,\big|\det\operatorname{Hess}H(\mathbf s)\big| \\
     &\hspace{2pc}\times\delta\big(NE-H(\mathbf s)\big)\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s)\big)
   \end{aligned}
 \end{equation}
 with two additional $\delta$-functions inserted to fix the energy density $E$
-and the stability $\mu$. The complexity is then
+and the stability $\mu$. The additional `noise' field $\boldsymbol\xi$
+helps regularize the $\delta$-functions for the energy and stability at finite
+$N$, and will be convenient for defining the order parameter matrices later. 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}\lim_{\sigma\to0}\frac1N\overline{\log\mathcal N(E, \mu}).
 \end{equation}
 Most of the difficulty of this calculation resides in the logarithm in this
 formula.
@@ -195,30 +197,51 @@ treat the logarithm inside the average of \eqref{eq:complexity}, and the
 $\delta$-functions are written in a Fourier basis. The average of the factor
 including the determinant and the factors involving $\delta$-functions can be
 averaged over the disorder separately \cite{Bray_2007_Statistics}. The result
-can be written as a function of three matrices indexed by the replicas: one
-which is a clear analogue of the usual overlap matrix of the equilibrium case,
-and two which can be related to the response of stationary points to
-perturbations of the potential. The general expression for the complexity as a
-function of these matrices is also found in \cite{Folena_2020_Rethinking}.
-
-We make the \emph{ansatz} that all three matrices have a hierarchical
-structure, and moreover that they share the same hierarchical structure. This
-means that the size of the blocks of equal value of each is the same, though
-the values inside these blocks will vary from matrix to matrix. This form can
-be shown to exactly reproduce the ground state energy predicted by the
-equilibrium solution, a key consistency check.
+can be written
+\begin{equation}
+  \Sigma(E,\mu)=\lim_{N\to\infty}\lim_{n\to0}\frac1N\frac{\partial}{\partial n}
+  \int_{\mathrm M_n(\mathbb R)} dQ\,dR\,dD\,e^{N\mathcal S(Q,R,D\mid E,\mu)},
+\end{equation}
+where the effective action $\mathcal S$ is a function of three matrices indexed
+by the $n$ replicas:
+\begin{equation}
+  \begin{aligned}
+    &Q_{ab}=\frac{\mathbf s_a\cdot\mathbf s_b}N
+    \hspace{4em}
+    R_{ab}=\frac{\boldsymbol\xi_a\cdot\mathbf s_b}{N\sigma^2}
+    \\
+    &D_{ab}=\frac1{N\sigma^4}\left(\sigma^2\delta_{ab}-\boldsymbol\xi_a\cdot\boldsymbol\xi_b\right).
+  \end{aligned}
+\end{equation}
+The matrix $Q$ is a clear analogue of the usual overlap matrix of the
+equilibrium case. The matrices $R$ and $D$ have interpretations as response
+functions: $R$ is related to the typical displacement of stationary points by
+perturbations to the potential, and $D$ is related to the change in the
+complexity caused by the same perturbations. The general expression for the
+complexity as a function of these matrices is also found in
+\cite{Folena_2020_Rethinking}.
+
+The complexity is found by the saddle point method, extremizing $\mathcal S$
+with respect to $Q$, $R$, and $D$ and replacing the integral with its integrand
+evaluated at the extremum. We make the \emph{ansatz} that all three matrices have
+a hierarchical structure, and moreover that they share the same hierarchical
+structure. This means that the size of the blocks of equal value of each is the
+same, though the values inside these blocks will vary from matrix to matrix.
+This form can be shown to exactly reproduce the ground state energy predicted
+by the equilibrium solution, a key consistency check.
 
 Along one line in the energy--stability plane the solution takes a simple form:
-the two hierarchical matrices corresponding to responses are diagonal, leaving
-only the overlap matrix with nontrivial off-diagonal entries. This
+the matrices $R$ and $D$ corresponding to responses are diagonal, leaving
+only the overlap matrix $Q$ with nontrivial off-diagonal entries. This
 simplification makes the solution along this line analytically tractable even
 for FRSB. The simplification is related to the presence of an approximate
 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.
+the TAP free energy \cite{Annibale_2003_Supersymmetric, Annibale_2003_The,
+Annibale_2004_Coexistence}. 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,
+overlap matrix $Q$ 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