Space-time correlations in the 1D Directed Stochastic Sandpile model

TL;DR

This paper derives recursive relations for space-time correlations in the 1D Directed Stochastic Sandpile, revealing positive correlations in particle density and negative correlations in avalanches, highlighting a balance between static and dynamic behaviors.

Contribution

It introduces recursive formulas for correlations in the 1D Directed Stochastic Sandpile, providing new insights into the interplay between static density and dynamic avalanches.

Findings

Particle density correlations are positive.

Avalanches are anticorrelated.

Balance between static and dynamic observables is governed by particle conservation.

Abstract

Sandpile models are known to resist exact results. In this direction, space-time correlations between avalanches have proven to be especially difficult to access. One of the main obstacle to do so comes from taking memory effects in a systematic way along the computation. In this paper, we partially fill this gap and derive recursive relations for the particle filling and avalanche 2-points correlation function in the 1D Directed Stochastic Sandpile. These expressions allow to characterize the sign of the correlations and estimates are provided in the particle filling case. In fact, density correlations are shown to be positively correlated. This behavior is directly related to persistence of the local particle filling. On the other hand, we show that avalanches are anticorrelated in the model. This is interpreted by the fact that avalanches disrupt the system and the damage can only be fully compensated after injecting a sufficiently high number of particles. These results indicate an underlying trade off, between static and dynamic observable, for the system to sit in its stationary state. It appears that this balance is controlled by the conservation of the particle number along the avalanches.