Convergence of equilibrium measures corresponding to finite subgraphs of infinite graphs: new examples

The paper establishes, for linear graphs, the existence of an irregular exhaustive subsequence G_{n_k,n_{k+1}} (Theorem 4.5) and, under the bounded-jumps assumption, the regularity of the canonical sequence G_n (Theorem 5.3), using a precise derivative criterion (Lemma 2.4) and careful control of the first-return generating functions ϕ_n, ϕ_{n,m} with extra cycle terms beyond simple truncations of ϕ (formulas (4.4)–(4.6)) . The candidate solution asserts the same conclusions but relies on two incorrect simplifications: (i) it identifies ϕ_{G_n} with S_n (omitting additional cycle lengths that appear in G_n) and (ii) it claims that, under bounded jumps, a fixed cylinder depends only on finitely many cycle lengths—both contradicted by the paper’s structure of Gn and Gn,m and the inevitability of additional cycles when cycles intersect .