Exact Solution of an One Dimensional Deterministic Sandpile Model

TL;DR

This paper provides an exact analytical solution for a one-dimensional deterministic sandpile model using the transfer matrix method, revealing detailed statistical properties and the multifractal nature of the system's limit cycle.

Contribution

It introduces an exact solution approach for the deterministic sandpile model for any system size, extending the transfer matrix method to analyze n-point functions and attractor structures.

Findings

Exact expressions for one- and two-point functions

Eigenvalues of the transfer matrix determine system correlations

Multifractal description of the limit cycle

Abstract

Using the transfer matrix method, we give the exact solution of a deterministic sandpile model for arbitrary $N$, where $N$ is the size of a single toppling. The one- and two-point functions are given in term of the eigenvalues of an $N \times N$ transfer matrix. All the n-point functions can be found in the same way. Application of this method to a more general class of models is discussed. We also present a quantitative description of the limit cycle (attractor) as a multifractal.