Pfaffian formulas for spanning tree probabilities

TL;DR

This paper presents a novel method to compute specific spanning tree probabilities in embedded graphs using Pfaffian formulas, involving advanced combinatorial structures and line-bundle Green's functions.

Contribution

It introduces Pfaffian formulas for spanning tree probabilities in annular graphs, linking combinatorics, topology, and linear algebra in a new way.

Findings

Pfaffian formulas accurately compute spanning tree probabilities.

Coefficients are related to cover-inclusive Dyck tilings.

Method applies to graphs embedded in an annulus.

Abstract

We show that certain topologically defined uniform spanning tree probabilities for graphs embedded in an annulus can be computed as linear combinations of Pfaffians of matrices involving the line-bundle Green's function, where the coefficients count cover-inclusive Dyck tilings of skew Young diagrams.