First-principle derivation of static avalanche-size distribution

TL;DR

This paper derives the static avalanche-size distribution for an elastic interface in a random potential using a first-principles approach, combining mean-field and renormalization group methods, and connects it to classical probability models.

Contribution

It introduces a systematic method to compute avalanche statistics from first principles, including 1-loop corrections and exact solutions at the upper critical dimension.

Findings

Derived the mean-field avalanche size distribution via saddle-point equations.

Computed 1-loop corrections using functional renormalization group techniques.

Established a connection between shock statistics and Levy processes, including the Brownian Force model.

Abstract

We study the energy minimization problem for an elastic interface in a random potential plus a quadratic well. As the position of the well is varied, the ground state undergoes jumps, called shocks or static avalanches. We introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements. The tree-level calculation, i.e. mean-field limit, is obtained by solving a saddle-point equation. Graphically, it can be interpreted as a the sum of all tree graphs. The 1-loop corrections are computed using results from the functional renormalization group. At the upper critical dimension the shock statistics is described by the Brownian Force model (BFM), the static version of the so-called ABBM model in the non-equilibrium context of depinning. This model can itself be treated exactly in any dimension and its shock statistics is that of a Levy process. Contact is made with classical results in probability theory on the Burgers equation with Brownian initial conditions. In particular we obtain a functional extension of an evolution equation introduced by Carraro and Duchon, which recursively constructs the tree diagrams in the field theory.