Manifolds in high dimensional random landscape: complexity of stationary points and depinning

TL;DR

This paper derives explicit formulas for the complexity of stationary points and minima in a high-dimensional random energy landscape, revealing phase transitions and bounds relevant to depinning phenomena.

Contribution

It provides the first explicit expressions for complexities in elastic manifolds in random media, linking them to phase transitions and depinning thresholds.

Findings

Complexities vanish at critical curvature $$ indicating phase transition.

Complexities exhibit quadratic and cubic vanishing near $$ for stationary points and minima.

Finite massless limit at zero curvature offers an upper bound for depinning threshold.

Abstract

We obtain explicit expressions for the annealed complexities associated respectively with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension $d<4$ embedded in a random medium of dimension $N \gg 1$ and confined by a parabolic potential with the curvature parameter $\mu$. These complexities are found to both vanish at the critical value $\mu_c$ identified as the Larkin mass. For $\mu<\mu_c$ the system is in complex phase corresponding to the replica symmetry breaking in its $T=0$ thermodynamics. The complexities vanish respectively quadratically (stationary points) and cubically (minima) at $\mu_c^-$. For $d\geq 1$ they admit a finite "massless" limit $\mu=0$ which is used to provide an upper bound for the depinning threshold under an applied force.