The Ihara Zeta function for infinite graphs

TL;DR

This paper introduces a unified framework for defining the Ihara Zeta function on measure graphs, encompassing finite, infinite, and percolation graphs, and provides a determinant formula and convergence results.

Contribution

It develops a general theory of measure graphs with groupoid actions, constructs a Zeta function using non-commutative integration, and extends the Ihara Zeta function to new classes of graphs.

Findings

Constructed a Zeta function for measure graphs using non-commutative integration.

Established a determinant formula for the Zeta function.

Showed compatibility with weak convergence and approximation by finite graphs.

Abstract

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and percolation graphs. Making use of Connes' non-commutative integration theory we construct a Zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara Zeta function via the normalized version on finite graphs in the sense of Benjamini-Schramm follows as a special case. Our framework not only unifies corresponding earlier results occurring in the literature. It likewise provides extensions to rich new classes of objects such as percolation graphs.