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diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index b6e7b8b..5c29523 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -26,17 +26,16 @@ \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} \begin{abstract} - {\color{red} Complex landscapes are defined as those having a proliferation of saddle points. - The question of their number and organization has been the object of long-standing attention, in particular centered around Gaussian mean-field potentials, - which includes glass and spin glass models. -The annealed approximation is by now well understood, but is exact for a restricted subset of these problems. Here we derive the exact quenched -solution for the general case, which incorporates Parisi's solution for the ground state, - as it should. More importantly, including - replica symmetry breaking uncovers the full distribution of saddles at given energy in terms of their stabilities, a structure that is lost in the annealed approximation. This structure should be a guide for the identification - of relevant activated processes in relaxational or driven dynamics.} - %These examples demonstrate the consistency of the - %solution and reveal that the signature of replica symmetry breaking at high - %energy densities is found in high-index saddles, not minima. + Complex landscapes are defined by their many saddle points. Determining their + number and organization is a long-standing problem, in particular for + tractable Gaussian mean-field potentials, which include glass and spin glass + models. The annealed approximation is well understood, but is generally not exact. Here we derive the exact + quenched solution for the general case, which incorporates Parisi's solution + for the ground state, as it should. More importantly, the quenched solution + correctly uncovers the full distribution of saddles at a given energy, a + structure that is lost in the annealed approximation. This structure should + be a guide for the accurate identification of the relevant activated + processes in relaxational or driven dynamics. \end{abstract} \maketitle @@ -48,49 +47,47 @@ size of the system \cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. Though they are often called `rough landscapes' to evoke the intuitive image of many minima in something like a mountain range, the metaphor to topographical landscapes is strained by the reality that these -complex landscapes exist in very high dimensions. -Many interesting versions of the problem have been treated, and the subject has -evolved into an active field of probability -theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity, -BenArous_2019_Geometry} and has been applied to -energy functions inspired by molecular biology, evolution, and machine learning -\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. - - - - - -The computation of the number of metastable states in this setting -was pioneered forty years ago by Bray and Moore -\cite{Bray_1980_Metastable}, who proposed the first calculation for the -Sherrington--Kirkpatrick model, in one of the -first applications of any replica symmetry breaking (RSB) scheme. As was clear -from the later results by Parisi \cite{Parisi_1979_Infinite}, their result was not exact, and the problem has -been open ever since. To date the program of -computing the statistics of stationary points---minima, saddle points, and -maxima---of mean-field complex landscapes has been only carried out in an exact form for a relatively small subset of -models, including most notably the (pure) $p$-spin model ($p>2$) -\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary}. +complex landscapes exist in very high dimensions. Many interesting versions +of the problem have been treated, and the subject has evolved into an active +field of probability theory \cite{Auffinger_2012_Random, + Auffinger_2013_Complexity, BenArous_2019_Geometry} and has been applied to + energy functions inspired by molecular biology, evolution, and machine + learning \cite{Maillard_2020_Landscape, Ros_2019_Complex, + Altieri_2021_Properties}. + +The computation of the number of metastable states in such a landscape was +pioneered forty years ago by Bray and Moore \cite{Bray_1980_Metastable} on the +Sherrington--Kirkpatrick (SK) model in one of the first applications of any +replica symmetry breaking (RSB) scheme. As was clear from the later results by +Parisi \cite{Parisi_1979_Infinite}, their result was not exact, and the +problem has been open ever since. To date the program of computing the +statistics of stationary points---minima, saddle points, and maxima---of +mean-field complex landscapes has been only carried out in an exact form for a +relatively small subset of models, including most notably the (pure) $p$-spin spherical +model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, +Cavagna_1997_An, Cavagna_1998_Stationary}. {\color{red} -Having a full, exact (`quenched') solution of the generic problem is not -primarily a matter of {\em accuracy}. -Very basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given -energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or --- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? -These questions need to be answered for the understanding of the relevance of more complex objects such as -barrier crossing (which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics -(which end in what kind of target states?). - - - - - +Having a full, exact (`quenched') solution of the generic problem is not +primarily a matter of {\em accuracy}. Very basic structural questions are +omitted in the approximate `annealed' solution. What is lost is the nature, +at any given energy (or free energy) level, of the stationary points in a +generic energy function: at low energies are they basically all minima, with an +exponentially small number of saddles, or -- as we show here -- do they consist +of a mixture of saddles whose index -- the number of unstable directions -- is +a smoothly distributed number? These questions need to be answered for the +understanding of the relevance of more complex objects such as barrier crossing +(which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate +of long-time dynamics (which end in what kind of target states?). In fact, we show that the +state of dynamics in generic cases is limited to energies \emph{at which saddles +are exponentially more numerous than minima}. In this paper we present what we argue is the general replica ansatz for the number of stationary points of generic mean-field models, which we expect to -include the Sherrington--Kirkpatrick model. This allows us -to clarify the rich structure of all the saddles, and in particular the lowest ones. The interpretation of a Parisi ansatz itself, in this context must be recast in a way that makes sense for the order parameters involved. +include the SK model. This allows us to clarify the rich structure of all the +saddles, and in particular the lowest ones. The interpretation of a Parisi +ansatz itself, in this context must be recast in a way that makes sense for the +order parameters involved. } @@ -108,25 +105,20 @@ complications arise. In this paper and its longer companion, we share the first results for the complexity with nontrivial hierarchy \cite{Kent-Dobias_2022_How}. Using a general form for the solution detailed in a companion article, we describe the -structure of 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.}. - -For definiteness, we consider the standard example of 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} +structure of landscapes with a 1RSB complexity and a full RSB complexity. + +For definiteness, we consider the standard example of the mixed $p$-spin +spherical models, which exhibit a zoo of orders and phases. These models can be +defined by taking a random Gaussian Hamiltonian $H$ defined on the $N-1$ sphere +and with a covariance that depends on only the dot product (or overlap) between +two configurations. For $s_1,s_2\in S^{N-1}$, +\begin{equation} \label{eq:covariance} + \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right), \end{equation} - $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere -$\{|\mathbf s\|^2=N\}$. 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 -constants that define the particular model. For instance, the so-called `pure' -models have $a_p=1$ for some $p$ and all others zero. +where $f$ is a function with positive coefficients. This uniquely defines the +distribution over Hamiltonians $H$. The overbar will always denote an average +over the functions $H$. The choice of function $f$ fixes the model. For +instance, the `pure' $p$-spin models have $f(q)=\frac12q^p$. The complexity of the $p$-spin models has been extensively studied by physicists and mathematicians. Among physicists, the bulk of work has been on @@ -139,21 +131,6 @@ complexity have been used to study many geometric properties of the pure models, from the relative position of stationary points to one another to shape and prevalence of instantons \cite{Ros_2019_Complexity, Ros_2021_Dynamical}. -{\color{green} {\bf eliminate?} The variance of the couplings implies that the covariance of the energy with -itself depends on only the dot product (or overlap) between two configurations. -In particular, one finds -\begin{equation} \label{eq:covariance} - \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right), -\end{equation} -where $f$ is defined by the series -\begin{equation} - f(q)=\frac12\sum_pa_pq^p. -\end{equation} -One needn't start with a Hamiltonian like -\eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule -\eqref{eq:covariance} can be specified for arbitrary, non-polynomial $f$, as in -the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}. In fact, defined this way the mixed spherical model encompasses all isotropic Gaussian fields on the sphere.} - The family of spherical models thus defined is quite rich, and by varying the covariance $f$ nearly any hierarchical structure can be found in equilibrium. Because of a correspondence between the ground state complexity |