Minimal configurations and sandpile measures

TL;DR

This paper introduces a new simple method to construct the sandpile measure on infinite graphs, providing explicit formulas for minimal configurations and extending the understanding of their limiting probabilities.

Contribution

It presents a novel construction of the sandpile measure under minimal assumptions and derives determinantal formulas for minimal configurations on general graphs.

Findings

Constructed the sandpile measure with minimal assumptions.

Derived determinantal formulas for minimal configurations.

Extended the existence of limiting probabilities to generalized minimal configurations.

Abstract

We give a new simple construction of the sandpile measure on an infinite graph G, under the sole assumption that each tree in the Wired Uniform Spanning Forest on G has one end almost surely. For, the so called, generalized minimal configurations the limiting probability on G exists even without this assumption. We also give determinantal formulas for minimal configurations on general graphs in terms of the transfer current matrix.