Manifolds in a high-dimensional random landscape: Complexity of stationary points and depinning

Yan V. Fyodorov; Pierre Le Doussal

doi:10.1103/PhysRevE.101.020101

Manifolds in a high-dimensional random landscape: Complexity of stationary points and depinning

Yan V. Fyodorov, Pierre Le Doussal

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

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≫1 and confined by a parabolic potential with the curvature parameter μ. These complexities are found to both vanish at the critical value μc identified as the Larkin mass. For μ<μc 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 μc-. For d≥1 they admit a finite "massless" limit μ=0 which is used to provide an upper bound for the depinning threshold under an applied force.

Original language	English
Article number	020101(R)
Journal	PHYSICAL REVIEW E
Volume	101
Issue number	2
DOIs	https://doi.org/10.1103/PhysRevE.101.020101
Publication status	Published - 18 Feb 2020

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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≫1 and confined by a parabolic potential with the curvature parameter μ. These complexities are found to both vanish at the critical value μc identified as the Larkin mass. For μ<μc 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 μc-. For d≥1 they admit a finite {"}massless{"} limit μ=0 which is used to provide an upper bound for the depinning threshold under an applied force.",

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N2 - 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≫1 and confined by a parabolic potential with the curvature parameter μ. These complexities are found to both vanish at the critical value μc identified as the Larkin mass. For μ<μc 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 μc-. For d≥1 they admit a finite "massless" limit μ=0 which is used to provide an upper bound for the depinning threshold under an applied force.

AB - 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≫1 and confined by a parabolic potential with the curvature parameter μ. These complexities are found to both vanish at the critical value μc identified as the Larkin mass. For μ<μc 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 μc-. For d≥1 they admit a finite "massless" limit μ=0 which is used to provide an upper bound for the depinning threshold under an applied force.

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Manifolds in a high-dimensional random landscape: Complexity of stationary points and depinning

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