Five shades of KMS: Statistical properties in the spectral geometry of Cuntz--Krieger algebras

We study spectral invariants arising in the noncommutative geometry of topological Markov chains and Cuntz–Krieger algebras. Their noncommutative geometry is described by spectral triples built from log-Laplacians, which are known to have non-trivial index theory and exotic quantum symmetries. We prove statistical eigenvalue asymptotics, local heat trace asymptotics, local Weyl laws, and an analogue of Connes’ trace theorem. In all cases the local asymptotics are governed by the Kubo–Martin–Schwinger state of the gauge action on the Cuntz–Krieger algebra.