Algebraic Limits of Sandpiles

TL;DR

This paper develops algebraic foundations for sandpile models, exploring their limiting structures and revealing new divisibility properties related to spanning trees on rectangles.

Contribution

It introduces a novel algebraic framework for sandpiles, connecting limits of extended and classical sandpile groups through canonical epimorphisms.

Findings

Identifies the limiting structure of the extended sandpile group.

Establishes relationships between different sandpile group limits.

Discovers divisibility properties of spanning trees on rectangles.

Abstract

The paper contributes to building algebraic foundations of self-organized criticality answering a previously unsolved question about the limiting structure of the extended sandpile group as well as relating it to another limit at the level of classical sandpile groups with respect to certain monomorphisms, and puts forward a concept of canonical sandpile epimorphisms, drawing an unexpected consequence about the divisibility properties of the numbers of spanning trees on rectangles.