Avalanche polynomials

TL;DR

This paper investigates the properties of avalanche polynomials on plane trees within the abelian sandpile model, demonstrating computational complexity results and deriving formulas for distribution statistics.

Contribution

It proves that constructing a tree with a given avalanche polynomial is NP-complete and provides formulas for the average and variance of avalanche distributions on trees.

Findings

NP-completeness of the prescribed polynomial problem

Closed-form formulas for average and variance of avalanche distributions

Analysis of avalanche polynomial properties on plane trees

Abstract

The avalanche polynomial on a graph captures the distribution of avalanches in the abelian sandpile model. Studied on trees, this polynomial could be defined by simply considering the size of the subtrees of the original tree. In this article, we study some properties of this polynomial on plane trees. Previously it has been proved that two different trees could have the same avalanche polynomial. We show here that the problem of finding a tree with a prescribed polynomial is NP-complete. In a second part, we study the average and the variance of the avalanche distribution on trees and give a closed formula.