Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts

The paper’s Theorem 30 proves the finiteness/infinity criterion for eigenmeasures on the renewal shift via a one-step inequality (5.6)–(5.8) and defines the critical inverse temperature β̃c by the limsup condition, adding that if Var1φ<∞ then νβ is finite for all β∈(0,βc) and, equivalently, β̃c=βc with an alternative expression using φn(xn)/n . The candidate solution derives essentially the same criterion using a conformality identity on injective branches and a bounded-distortion estimate that avoids Var1φ, and it also shows β̃c=βc under Var1φ<∞. The main technical slip in the model is the claim that ΣA\[1] is a disjoint union of the cylinders [γjn]; this family is not disjoint across different n, but the argument only needs an upper bound and goes through with a union bound. Aside from this repairable point and some unstated hypotheses (e.g., supφ<∞), the model’s proof matches the paper’s conclusions and uses a different, but standard, renewal/inducing viewpoint (via Kac’s formula) instead of the paper’s appeal to Sarig’s discriminant theorem. Overall, both reach the same result by closely related but not identical routes .