Large deviations of avalanches in the raise and peel model

TL;DR

This paper investigates the large deviation properties of avalanche dynamics in the Raise and Peel model, linking it to the XXZ model, and analyzes phase transitions in the rate functions of global avalanches.

Contribution

It introduces a novel connection between avalanche statistics in the Raise and Peel model and the groundstate eigenvalues of the XXZ model, providing asymptotic cumulant evaluations.

Findings

Confirmed exact formulas for mean avalanche quantities.

Identified phase transition in the rate function for global avalanches.

Analyzed asymptotic behavior of large system size deviations.

Abstract

We study the large deviation functions for two quantities characterizing the avalanche dynamics in the Raise and Peel model: the number of tiles removed by avalanches and the number of global avalanches extending through the whole system. To this end, we exploit their connection to the groundstate eigenvalue of the XXZ model with twisted boundary conditions. We evaluate the cumulants of the two quantities asymptotically in the limit of the large system size. The first cumulants, the means, confirm the exact formulas conjectured from analysis of finite systems. We discuss the phase transition from critical to non-critical behaviour in the rate function of the global avalanches conditioned to an atypical values of the number of tiles removed by avalanches per unit time.