Sandpiles with height restrictions

TL;DR

This paper investigates a simplified one- and two-dimensional stochastic sandpile model with a height restriction, analyzing its phase transition behavior and critical exponents through simulations and cluster approximations.

Contribution

It introduces a height-restricted sandpile model that simplifies analysis while preserving critical behavior, comparing two toppling rules and confirming universality.

Findings

The model exhibits a continuous phase transition at a critical density.

Critical exponents match those of the unrestricted Manna sandpile.

Height restriction simplifies state space without altering universality class.

Abstract

We study stochastic sandpile models with a height restriction in one and two dimensions. A site can topple if it has a height of two, as in Manna's model, but, in contrast to previously studied sandpiles, here the height (or number of particles per site), cannot exceed two. This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded. Two toppling rules are considered: in one, the particles are redistributed independently, while the other involves some cooperativity. We study the fixed-energy system (no input or loss of particles) using cluster approximations and extensive simulations, and find that it exhibits a continuous phase transition to an absorbing state at a critical value zeta_c of the particle density. The critical exponents agree with those of the unrestricted Manna sandpile.