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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 913a77e..3ee3e26 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -80,6 +80,20 @@ way. One of the purposes of this paper is to give a sufficiently detailed characterization of a general landscape so that a meaningful general notion of threshold may be introduced - if this is at all possible. +The format of this paper is as follows. In \S\ref{sec:model}, we introduce the +mean-field model we study, the mixed $p$-spin spherical model. In +\S\ref{sec:equilibrium} we review details of the equilibrium solution that will +be relevant in our study of the landscape complexity. In \S\ref{sec:complexity} +we derive a generic form for the complexity of the model. In \S\ref{sec:ansatz} +we make and review the hierarchical replica symmetry breaking ansatz used to +solve the complexity. In \S\ref{sec:supersymmetric} we write down the solution +in a specific and limited regime, which is nonetheless helpful as it gives a +foothold for numerically computing the complexity everywhere else. +\S\ref{sec:frsb} explains aspects of the solution specific to the case of full +RSB, and derives the RS--FRSB transition line. \S\ref{sec:examples} details the +landscape topology of two example models: a $3+16$ model with a 2RSB ground +state, and a $2+4$ with a FRSB ground state. Finally \S\ref{sec:interpretation} +provides some interpretation of our results. \section{The model} \label{sec:model} @@ -129,6 +143,7 @@ its number of negative eigenvalues, the index $\mathcal I$ of the saddle (see Fyodorov \cite{Fyodorov_2007_Replica} for a detailed discussion). \section{Equilibrium} +\label{sec:equilibrium} Here we review the equilibrium solution \cite{Crisanti_1992_The, Crisanti_1993_The, Crisanti_2004_Spherical, Crisanti_2006_Spherical}. The free @@ -218,6 +233,7 @@ Kac--Rice computation is also given by a $(k-1)$-RSB anstaz. Heuristically, this each stationary point also has no width and therefore overlap one with itself. \section{Landscape complexity} +\label{sec:complexity} The stationary points of a function can be counted using the Kac--Rice formula, which integrates a over the function's domain a $\delta$-function containing @@ -493,6 +509,7 @@ saddle densities. \section{Replica ansatz} +\label{sec:ansatz} Based on previous work on the Sherrington--Kirkpatrick model and the equilibrium solution of the spherical model, we expect $C$, and $R$ and $D$ to @@ -566,6 +583,7 @@ functions $c(x)$, $r(x)$ and $d(x)$, then carrying through the appropriate operations. \section{Supersymmetric solution} +\label{sec:supersymmetric} The Kac--Rice problem has an approximate supersymmetry, which is found when the absolute value of the determinant is neglected, which has been studied in great detail in the complexity of the Thouless--Anderson--Palmer free energy \cite{Annibale_2003_The, Annibale_2003_Supersymmetric, Annibale_2004_Coexistence}. When this is done, the @@ -668,6 +686,7 @@ sufficiently close to the correct answer. This is the strategy we use in \S\ref{sec:examples}. \section{Full replica symmetry breaking} +\label{sec:frsb} This reasoning applies equally well to FRSB systems. \begin{align} |