Asymptotic Pressure on the Cayley Graph of a Finitely Generated Semigroup

The paper’s main theorem (Theorem 3.2) asserts that for primitive/irreducible tree restrictions R_k with Perron–Frobenius eigenvalues λ_k → ∞ and a fixed interaction matrix E(i,j) = a_{ij} w_j with maximal row sum s, the asymptotic pressure satisfies lim_{k→∞} P(k) = log s. The setup, partition function, and pressure definitions are standard in the paper’s Section 2 (E, Z_n, P_n) , and Lemma 3.1 gives the needed layer-size ratio |L_n|/|Δ_n| → (λ_k−1)/λ_k . The upper bound P(k) ≤ log s follows from the “leaf-peeling” estimate (3.10)–(3.11) . However, the paper’s lower bound step (3.12)–(3.14) is incomplete: it selects a label i with ∑_j E(i,j)=s and claims Z_n ≥ s^{|L_n|} min_i w_i without accounting for the contributions of all edges above the last layer and without justifying the existence of predecessors of i on L_{n-2} (and higher) for arbitrary A; this glosses over missing positive lower bounds for those contributions and an implicit reachability assumption for i . In contrast, the candidate solution supplies the missing ingredients: (i) a uniform positive lower bound using w_* and E_* on earlier edges, which contributes only O(1/λ_k) in the final per-site limit, and (ii) a reachability condition (or an r-step grouping workaround) to ensure that bottom-layer resummation produces the s^{|L_n|} factor even when some maximizers are not one-step reachable. This yields P(k) ≥ ((λ_k−1)/λ_k) log s + O(1/λ_k) and hence lim_{k→∞} P(k) = log s, fully reconciling the asymptotic claim. Therefore, the result is right, but the paper’s proof omits essential assumptions/estimates; the model’s solution is correct and more complete.