Logarithmic scaling for height variables in the Abelian sandpile model

TL;DR

This paper provides an exact analysis of the scaling behavior of height variables in the 2D Abelian sandpile model, revealing their connection to a logarithmic conformal field theory and confirming results through simulations.

Contribution

It identifies the height 2 variable as a logarithmic scalar field of dimension 2 in the c=-2 conformal field theory, extending the analysis to other heights and proposing explicit 2-point functions.

Findings

Height 2 variable is a logarithmic scalar field of dimension 2.

Height 3 and 4 variables share the same scaling form as height 2.

Proposed bulk 2-point functions for all height variables.

Abstract

We report on the exact computation of the scaling form of the 1-point function, on the upper-half plane, of the height 2 variable in the two-dimensional Abelian sandpile model. By comparing the open versus the closed boundary condition, we find that the scaling field associated to the height 2 is a logarithmic scalar field of scaling dimension 2, belonging to a c=-2 logarithmic conformal field theory. This identification is confirmed by numerical simulations and extended to the height 3 and 4 variables, which exhibit the same scaling form. Using the conformal setting, we make precise proposals for the bulk 2-point functions of all height variables.