Mechanisms of evolution of avalanches in regular graphs

TL;DR

This paper establishes a mapping between avalanches in the zero-temperature random-field Ising model on regular graphs and population life-periods, providing criteria for infinite avalanches and analyzing their properties.

Contribution

It introduces a novel mapping and analytical framework to understand the conditions and characteristics of infinite avalanches in regular graphs.

Findings

Infinite avalanches occur only for q>3 in q-regular graphs.

Long avalanches show a significant degree of universality.

Analytical expressions for avalanche duration, pulse-shapes, and power spectrum are derived.

Abstract

A mapping of avalanches occurring in the zero-temperature random-field Ising model (zt-RFIM) to life-periods of a population experiencing immigration is established. Such a mapping allows the microscopic criteria for occurrence of an infinite avalanche in a q-regular graph to be determined. A key factor for an avalanche of spin flips to become infinite is that it interacts in an optimal way with previously flipped spins. Based on these criteria, we explain why an infinite avalanche can occur in q-regular graphs only for q>3, and suggest that this criterion might be relevant for other systems. The generating function techniques developed for branching processes are applied to obtain analytical expressions for the duration, pulse-shapes and power spectrum of the avalanches. The results show that only very long avalanches exhibit a significant degree of universality.