The four height variables, boundary correlations, and dissipative defects in the Abelian sandpile model

TL;DR

This paper investigates the field identifications and correlations of height variables in the 2D Abelian sandpile model, revealing boundary-dependent behaviors and the effects of dissipative defects, with implications for simplifying future calculations.

Contribution

It provides a detailed analysis of height variable correlations, boundary effects, and dissipative defects, proposing a unified field operator representation along open boundaries.

Findings

Height variables differ in bulk and closed boundaries but unify along open boundaries.

Dissipative defects do not affect correlations in the bulk or closed boundaries.

All heights and defects are represented by the same field operator along open boundaries.

Abstract

We analyze the two-dimensional Abelian sandpile model, and demonstrate that the four height variables have different field identifications in the bulk, and along closed boundaries, but become identical, up to rescaling, along open boundaries. We consider two-point boundary correlations in detail, and discuss a number of complications that arise in the mapping from sandpile correlations to spanning tree correlations; the structure of our results suggests a conjecture that could greatly simplify future calculations. We find a number of three-point functions along closed boundaries, and propose closed boundary field identifications for the height variables. We analyze the effects of dissipative defect sites, at which the number of grains is not conserved, and show that dissipative defects along closed boundaries, and in the bulk, have no effect on any weakly allowed cluster variables, or on their correlations. Along open boundaries, we find a particularly simple field structure; we calculate all $n$-point correlations, for any combinations of height variables and dissipative defect sites, and find that all heights and defects are represented by the same field operator.