Scaling limit for a long-range divisible sandpile

TL;DR

This paper investigates the scaling limit of a divisible sandpile model linked to a truncated alpha-stable random walk, revealing its connection to an obstacle problem for a truncated fractional Laplacian and providing precise asymptotics for Green's functions.

Contribution

It introduces the scaling limit of a long-range divisible sandpile model and establishes its relation to a truncated fractional Laplacian obstacle problem, with detailed Green's function asymptotics.

Findings

Limiting distribution related to a truncated fractional Laplacian obstacle problem

Asymptotic expansions for rescaled discrete Green's functions

Derived convergence rates of Green's functions to continuous counterparts

Abstract

We study the scaling limit of a divisible sandpile model associated to a truncated $\alpha$-stable random walk. We prove that the limiting distribution is related to an obstacle problem for a truncated fractional Laplacian. We also provide, as a fundamental tool, precise asymptotic expansions for the corresponding rescaled discrete Green's functions. In particular, the convergence rate of these Green's functions to its continuous counterpart is derived.