Pfaffians and nonintersecting paths in graphs with cycles: Grassmann algebra methods

TL;DR

This paper extends Pfaffian and nonintersecting path formulas to graphs with cycles using Grassmann algebra, generalizing classical results like Lindström-Gessel-Viennot.

Contribution

It introduces Grassmann algebra methods to generalize path and Pfaffian formulas to cyclic graphs, expanding their applicability.

Findings

Derived Pfaffian expressions for graphs with cycles.

Generalized Lindström-Gessel-Viennot formula.

Connected path systems with nonintersecting cycles.

Abstract

After recalling the definition of Grassmann algebra and elements of Grassmann--Berezin calculus, we use the expression of Pfaffians as Grassmann integrals to generalize a series of formulas relating generating functions of paths in digraphs to Pfaffians. We start with the celebrated Lindstr\"om-Gessel-Viennot formula, which we derive in the general case of a graph with cycles. We then make further use of Grassmann algebraic tools to prove a generalization of the results of (Stembridge 1990). Our results, which are applicable to graphs with cycles, are formulated in terms of systems of nonintersecting paths and nonintersecting cycles in digraphs.