diff options
| -rw-r--r-- | frsb_kac-rice_letter.tex | 89 |
1 files changed, 40 insertions, 49 deletions
diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index 5c29523..c7ef913 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -29,13 +29,14 @@ 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. + models. The annealed approximation is well understood, but is generally not + exact. Here we describe 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 @@ -67,58 +68,48 @@ relatively small subset of models, including most notably the (pure) $p$-spin sp 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?). 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}. +primarily a matter of {\em accuracy}. Basic structural questions are +omitted in the approximate `annealed' solution. What is lost is the nature of +the stationary points at a given energy level: 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 if one hopes to correctly describe more complex +objects such as barrier crossing (which barriers?) \cite{Ros_2019_Complexity, +Ros_2021_Dynamical} or the fate of long-time dynamics (that end in which kind +of states?). 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 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. - -} - -{\color{blue} - For simplicity we have concentrated here on the energy, rather -than {\em free-energy} landscape, although the latter is sometimes -more appropriate. From the technical point of view, this makes no fundamental difference, it suffices -to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. We do not expect new features or technical -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. +saddles, and in particular the lowest ones. Using a +general form for the solution detailed in a companion article \cite{Kent-Dobias_2022_How}, we describe the +structure of landscapes with a 1RSB complexity and a full RSB complexity. The interpretation of a Parisi +ansatz itself must be recast to make sense of the new order parameters +involved. + +For simplicity we concentrate on the energy, rather than {\em +free-energy}, landscape, although the latter is sometimes more appropriate. From +the technical point of view, this makes no fundamental difference and it suffices +to apply the same computation to the Thouless--Anderson--Palmer +\cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the +energy. We do not expect new features or technical complications to arise. 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 $s_1,s_2\in S^{N-1}$, +spherical models, which exhibit a zoo of disordered phases. These models can be +defined by drawing a random Hamiltonian $H$ from a distribution of isotropic +Gaussian fields defined on the $N-1$ sphere. Isotropy implies that the +covariance in energies between two configurations depends on only their dot +product (or overlap), so for $\mathbf s_1,\mathbf s_2\in +S^{N-1}$, \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} -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 $f(q)=\frac12q^p$. +where $f$ is a function with positive coefficients. The overbar will always +denote an average over the functions $H$. The choice of function $f$ uniquely +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 |