# Gaps between avalanches in 1D Random Field Ising Models

## TL;DR

This paper investigates the statistical distribution of gaps between avalanches in one-dimensional random field Ising models, revealing different behaviors in ferromagnetic and antiferromagnetic cases, with analytical and numerical results.

## Contribution

It provides the first analytical computation of avalanche gap distributions in 1D RFIMs and explores the effects of disorder and long-range interactions on these distributions.

## Key findings

- Gap distribution tends to a constant as gap size approaches zero in ferromagnetic RFIM.
- Antiferromagnetic RFIM exhibits a gapped behavior with a power-law onset near the offset.
- Numerical simulations confirm analytical predictions and estimate the exponent θ ≈ 0.95.

## Abstract

We analyze the statistics of gaps ($\Delta H$) between successive avalanches in one dimensional random field Ising models (RFIMs) in an external field $H$ at zero temperature. In the first part of the paper we study the nearest-neighbour ferromagnetic RFIM. We map the sequence of avalanches in this system to a non-homogeneous Poisson process with an $H$-dependent rate $\rho(H)$. We use this to analytically compute the distribution of gaps $P(\Delta H)$ between avalanches as the field is increased monotonically from $-\infty$ to $+\infty$. We show that $P(\Delta H)$ tends to a constant $\mathcal{C}(R)$ as $\Delta H \to 0^+$, which displays a non-trivial behaviour with the strength of disorder $R$. We verify our predictions with numerical simulations. In the second part of the paper, motivated by avalanche gap distributions in driven disordered amorphous solids, we study a long-range antiferromagnetic RFIM. This model displays a gapped behaviour $P(\Delta H) = 0$ up to a system size dependent offset value $\Delta H_{\text{off}}$, and $P(\Delta H) \sim (\Delta H - \Delta H_{\text{off}})^{\theta}$ as $\Delta H \to H_{\text{off}}^+$. We perform numerical simulations on this model and determine $\theta \approx 0.95(5)$. We also discuss mechanisms which would lead to a non-zero exponent $\theta$ for general spin models with quenched random fields.