Lots of writing.

1 files changed, 54 insertions, 22 deletions

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index d3369ac..801dc8e 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -45,13 +45,13 @@ complex landscapes also exist in very high dimensions: think of the dimensions
 of phase space for $N$ particles, or the number of parameters in a neural
 network.
 
-The \emph{complexity} of a function is the logarithm of the average number of
-its minima, maxima, and saddle points (collectively stationary points), under
-conditions like the value of the energy or the index of the stationary point.
-Since in complex landscapes this number grows exponentially with system size,
-their complexity is an extensive quantity. Understanding the complexity offers
-an understanding about the geometry and topology of the landscape, which can
-provide insight into dynamical behavior.
+The \emph{complexity} of a function is the average of the logarithm of the
+number of its minima, maxima, and saddle points (collectively stationary
+points), under conditions like the value of the energy or the index of the
+stationary point.  Since in complex landscapes this number grows exponentially
+with system size, their complexity is an extensive quantity. Understanding the
+complexity offers an understanding about the geometry and topology of the
+landscape, which can provide insight into dynamical behavior.
 
 When complex systems are fully connected, i.e., each degree of freedom
 interacts directly with every other, they are often described by a hierarchical
@@ -65,20 +65,20 @@ only been computed in RS cases.
 
 In this paper we share the first results for the complexity with nontrivial
 hierarchy. Using a general form for the solution, we detail the structure of
-landscapes with a 1RSB complexity and a full RSB complexity.
-
-The Thouless--Anderson--Palmer (TAP) complexity is the complexity of a kind of
-mean-field free energy. Because of some deep thermodynamic relationships
-between the TAP complexity and the equilibrium free energy, the TAP complexity
-can be computed with extensions of the equilibrium method. As a result, the TAP
-complexity has been previously computed for nontrivial hierarchical structure.
+landscapes with a 1RSB complexity and a full RSB complexity \footnote{The
+  Thouless--Anderson--Palmer (TAP) complexity is the complexity of a kind of
+  mean-field free energy. Because of some deep thermodynamic relationships
+  between the TAP complexity and the equilibrium free energy, the TAP
+  complexity can be computed with extensions of the equilibrium method. As a
+  result, the TAP complexity has been previously computed for nontrivial
+hierarchical structure.}.
 
 We study the mixed $p$-spin spherical models, with Hamiltonian
 \begin{equation} \label{eq:hamiltonian}
   H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
 \end{equation}
-is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the sphere
-$\|\mathbf s\|^2=N$.  The coupling coefficients $J$ are taken at random, with
+is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere
+$S^{N-1}=\{\mathbf s\mid\|\mathbf s\|^2=N\}$.  The coupling coefficients $J$ are taken at random, with
 zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that
 the energy is typically extensive. The overbar will always denote an average
 over the coefficients $J$. The factors $a_p$ in the variances are freely chosen
@@ -136,7 +136,7 @@ It's worth reviewing the complexity for the best-studied case of the pure model
 for $p\geq3$. Here, because the covariance is a homogeneous polynomial, $E$ and
 $\mu$ cannot be fixed separately, and one implies the other: $\mu=pE$.
 Therefore at each energy there is only one kind of stationary point. When the
-energy reaches $E_\mathrm{th}=\mu_m/p$, the population of stationary points
+energy reaches $E_\mathrm{th}=-\mu_m/p$, the population of stationary points
 suddenly shifts from all saddles to all minima, and there is an abrupt
 percolation transition in the topology of constant-energy slices of the
 landscape. This behavior of the complexity can be used to explain a rich
@@ -148,11 +148,42 @@ dynamics, and to the algorithmic limit $E_\mathrm{alg}$.
 Things become much less clear in even the simplest mixed models. For instance,
 one mixed model known to have a replica symmetric complexity was shown to
 nonetheless not have a clear relationship between features of the complexity
-and the asymptotic dynamics \cite{Folena_2020_Rethinking}.
+and the asymptotic dynamics \cite{Folena_2020_Rethinking}. There is no longer a
+sharp topological transition.
 
 To compute the complexity in the generic case, we use the replica method to
 treat the logarithm inside the average of \eqref{eq:complexity}, and the
-$\delta$-functions are written in a Fourier basis.
+$\delta$-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.
+
+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
+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.
+
+In general, solving the saddle-point equations for the parameters of the three
+replica matrices is challenging. Unlike the equilibrium case, the solution is
+not extremal, and so minimization methods cannot be used. However, the line of
+simple `supersymmetric' solutions offers a convenient foothold: starting from
+one of these solutions, the parameters $E$ and $\mu$ can be slowly varied to
+find the complexity everywhere. This is how the data in what follows was produced.
 
 \begin{figure}
   \centering
@@ -176,9 +207,10 @@ $\delta$-functions are written in a Fourier basis.
   } \label{fig:2rsb.phases}
 \end{figure}
 
-It is known that by choosing a covariance $f$ as the sum of polynomials with
-well-separated powers, one develops 2RSB in equilibrium. This should correspond
-to 1RSB in Kac--Rice. For this example, we take
+For the first example, we study a model whose complexity has the simplest
+replica symmetry breaking scheme, 1RSB. By choosing a covariance $f$ as the sum
+of polynomials with well-separated powers, one develops 2RSB in equilibrium.
+This should correspond to 1RSB in the complexity. For this example, we take
 \begin{equation}
   f(q)=\frac12\left(q^3+\frac1{16}q^{16}\right)
 \end{equation}