Spatial realisations of KMS states on the C*-algebras of higher-rank graphs

TL;DR

This paper characterizes the structure of KMS states at critical inverse temperature for higher-rank graph C*-algebras, extending known results from rank-one graphs to more complex higher-rank cases.

Contribution

It provides a new description of KMS states for higher-rank graph algebras under various dynamics, especially for rank-two graphs, using measures on mixed infinite-finite path spaces.

Findings

KMS states at critical temperature can be represented via measures on path spaces.

The results are most precise for rank-two graphs.

Extension of phase transition analysis to higher-rank graph algebras.

Abstract

Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a "preferred dynamics" for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by intregrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.