Exact integration of height probabilities in the Abelian Sandpile Model

TL;DR

This paper analytically computes the height probabilities in the Abelian Sandpile Model, confirming previous numerical and conjectured results, and providing exact integral evaluations to validate the model's predictions.

Contribution

It presents a direct analytical derivation of height probabilities in the Abelian Sandpile Model, confirming prior conjectures and numerical results.

Findings

Confirmed the conjectured cubic rational-coefficient polynomial form of probabilities

Validated the average height conjecture in the Abelian Sandpile Model

Provided exact integral evaluations of height probabilities

Abstract

The height probabilities for the recurrent configurations in the Abelian Sandpile Model on the square lattice have analytic expressions, in terms of multidimensional quadratures. At first, these quantities have been evaluated numerically with high accuracy, and conjectured to be certain cubic rational-coefficient polynomials in 1/pi. Later their values have been determined by different methods. We revert to the direct derivation of these probabilities, by computing analytically the corresponding integrals. Yet another time, we confirm the predictions on the probabilities, and thus, as a corollary, the conjecture on the average height.