# Correlations between avalanches in the depinning dynamics of elastic interfaces

## TL;DR

This paper develops a field theory to analyze correlations between avalanches in elastic interface depinning, revealing anti-correlations and proposing new scaling relations, with validation against simulations and implications for creep regimes.

## Contribution

It introduces a simple field theory capturing deviations from uncorrelated avalanches and predicts anti-correlations, advancing understanding of avalanche interactions in depinning dynamics.

## Key findings

- Avalanches are generally anti-correlated, with larger ones reducing the likelihood of subsequent large events.
- Derived new scaling relations for critical exponents governing avalanche correlations.
- Validated predictions through numerical simulations and compared with static avalanche results.

## Abstract

We study the correlations between avalanches in the depinning dynamics of elastic interfaces driven on a random substrate. In the mean field theory (the Brownian force model), it is known that the avalanches are uncorrelated. Here we obtain a simple field theory which describes the first deviations from this uncorrelated behavior in a $\epsilon=d_c-d$ expansion below the upper critical dimension $d_c$ of the model. We apply it to calculate the correlations between (i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or more generally, in two avalanches separated by a uniform displacement $W$ of the interface. For (i) we obtain the correlations of the total sizes, of the local sizes and of the total sizes with given seeds (starting points). For (ii) we obtain the correlations of the velocities, of the durations, and of the avalanche shapes. In general we find that the avalanches are {\it anti-correlated}, the occurence of a larger avalanche making more likely the occurence of a smaller one, and vice-versa. Examining the universality of our results leads us to conjecture several new exact scaling relations for the critical exponents that characterize the different distributions of correlations. The avalanche size predictions are confronted to numerical simulations for a $d=1$ interface with short range elasticity. They are also compared to our recent related work on static avalanches (shocks). Finally we show that the naive extrapolation of our result into the thermally activated creep regime at finite temperature, predicts strong positive correlations between the forward motion events, as recently observed in numerical simulations.