Exact Solution of an One Dimensional Deterministic Sandpile Model

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.