Sandpile solitons via smoothing of superharmonic functions

TL;DR

This paper develops a theory of smoothing superharmonic functions to prove the existence of solitons in a sandpile model, advancing understanding of their formation, interactions, and invariance properties.

Contribution

It introduces a new smoothing framework for superharmonic functions to establish the existence and properties of sandpile solitons.

Findings

Existence of minimal superharmonic functions matching given data at infinity.

Sandpile states with solitons are invariant under wave propagation.

Solitons can interact, forming triads and nodes.

Abstract

Let $F:\mathbb Z^2\to \mathbb Z$ be the pointwise minimum of several linear functions. The theory of smoothing of integer-valued superharmonic function allows us to prove that under certain conditions there exists the pointwise minimal superharmonic function which coincides with $F$ "at infinity". We develop such a theory to prove the existence of so-called solitons (or strings) in a certain sandpile model, studied by S. Caracciolo, G. Paoletti, and A. Sportiello. Thus we made a step towards understanding the phenomena of the identity in the sandpile group for a square where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we send waves (that is why we call them solitons), and can interact, forming triads and nodes.