Distribution of rare saddles in the p -spin energy landscape

Abstract

We compute the statistical distribution of index-1 saddles surrounding a given local minimum of the p-spin energy landscape, as a function of their distance to the minimum in configuration space and of the energy of the latter. We identify the saddles also in the region of configuration space in which they are subdominant in number (i.e., rare) with respect to local minima, by computing large deviation probabilities of the extremal eigenvalues of their Hessian. As an independent result, we determine the joint large deviation probability of the smallest eigenvalue and eigenvector of a GOE matrix perturbed with both an additive and multiplicative finite-rank perturbation.

Does Scrambling Equal Chaos?

“…is V j (q) = σ cos(q + β j ) where β j 's are uniformly distributed in [0, 2π] and σ > 0 is the disorder strength. In the strong disorder regime, such a system is known to have a complex "glassy" energy landscape, with an exponentially large number of equilibria with a wide range of energies [44][45][46][47]. Numerically (see caption of Fig.…”

Dynamical instantons and activated processes in mean-field glass models

“…We focus on the energy landscape associated to the p-spin spherical model: (1) has been the subject of an extensive amount of research devoted to understanding its statistical properties, which started with the earlier investigations [26][27][28][29][30][31] and culminated in the most recent results [22,23,32,33]. These works highlighted a peculiar organization of the landscape stationary points in terms of their energy density ε = E/N and of their stability: while at large value of the energy the landscape is dominated by saddles with a huge index (i.e., number of unstable directions), the local minima and low-index saddles concentrate at the bottom of the landscape, below a critical threshold value of the energy density ε th .…”

“…In particular, it is important to scan the landscape in the vicinity of any of its stationary points. This information is accessible via large deviation techniques by computing the complexity of the stationary points constrained to be at fixed, non-zero overlap from the reference stationary point [22][23][24]. In Ref.…”

“…We report in this subsection the expression of the terms S[s t ,ŝ t ] and K[s t ,ŝ t ] appearing in Eq. (23), that are the relevant ones for the derivation of the dynamical equations given below. The action S[s t ,ŝ t ] can be written as:…”

“…Here we provide the first computation of such dynamical instanton and characterize the properties of the new minima reached after barrier-crossing. In order to achieve this goal we make use of the results we obtained recently on the number of the stationary points constrained to be at fixed overlap q (or distance d) from a given minimum in a prototypical energy landscape [22,23] (see also [24] for a related analysis). These studies showed that, given an arbitrary local minimum s 1 of the energy landscape with energy density ε 1 , the landscape in its vicinity is populated by index-1 saddles, that constitute available escape states when the system is trapped in s 1 .…”