Asymptotic behavior of the order parameter in a stochastic sandpile

TL;DR

This paper derives a series expansion for the stationary activity density in a stochastic sandpile model, using a reorganized perturbation theory and extensive diagram enumeration, with predictions matching simulations.

Contribution

It presents the first four terms of a series expansion for the order parameter in a stochastic sandpile, utilizing a novel reorganization of perturbation theory and computational diagram enumeration.

Findings

Series expansion matches simulation results

Effective action isolates conserved particle density

Developed algorithm for diagram enumeration

Abstract

We derive the first four terms in a series for the order paramater (the stationary activity density rho) in the supercritical regime of a one-dimensional stochastic sandpile; in the two-dimensional case the first three terms are reported. We reorganize the pertubation theory for the model, recently derived using a path-integral formalism [R. Dickman e R. Vidigal, J. Phys. A 35, 7269 (2002)], to obtain an expansion for stationary properties. Since the process has a strictly conserved particle density p, the Fourier mode N^{-1} psi_{k=0} -> p, when the number of sites N -> infinity, and so is not a random variable. Isolating this mode, we obtain a new effective action leading to an expansion for rho in the parameter kappa = 1/(1+4p). This requires enumeration and numerical evaluation of more than 200 000 diagrams, for which task we develop a computational algorithm. Predictions derived from this series are in good accord with simulation results. We also discuss the nature of correlation functions and one-site reduced densities in the small-kappa (large-p) limit.