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

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