Boundary conditions and defect lines in the Abelian sandpile model

TL;DR

This paper investigates how introducing defect lines and varying boundary dissipation in the Abelian sandpile model affects its boundary conditions and universality class, revealing a universal coefficient for classifying boundary behaviors.

Contribution

It demonstrates that defect lines renormalize to boundary separations and that boundary dissipation variations do not change the universality class, introducing a universal coefficient for boundary classification.

Findings

Defect lines separate the plane into two half planes with open boundary conditions.

Varying boundary dissipation does not alter the universality class.

A universal coefficient classifies boundary conditions near defects.

Abstract

We add a defect line of dissipation, or crack, to the Abelian sandpile model. We find that the defect line renormalizes to separate the two-dimensional plane into two half planes with open boundary conditions. We also show that varying the amount of dissipation at a boundary of the Abelian sandpile model does not affect the universality class of the boundary condition. We demonstrate that a universal coefficient associated with height probabilities near the defect can be used to classify boundary conditions.