Potential theory and Zd-extensions

The paper’s Theorem 0.1 establishes, under ergodicity/recurrence and an irreducibility notion, the asymptotic equivalence between a first-order expansion of the finite-state transition matrices P_ε and of the discretized potential operators Q_ε, with the identification S = R_0^{-1} on the mean-zero subspace. Crucially, the paper does not claim that Q_ε = (Id − P_ε)^{-1} holds exactly for general Zd-extensions; that exact identity holds only for random walks (Markov chains). The candidate solution hinges on an exact resolvent identity (Id−P_ε)Q_ε = Id on C_I^0 derived from the claim ⟨f, Π^↑ L_ε^n Π_↓ g⟩ = ⟨f, P_ε^n g⟩ for all n. But beyond n=1 this fails because Π_*Π^* ≠ Id on L^1([I]) (the projection is not invariant under L_ε), so compression and powers/resolvents do not commute. The paper explicitly treats this subtlety, showing only asymptotic equivalence between (Id−Π^* L_ε Π_*)^{-1} and Π^*(Id−L_ε)^{-1}Π_* (Remark 1.16), and proves the two directions via nontrivial arguments (cone contraction, hitting-time estimates, bootstrap via the two-site case). Therefore, while the candidate’s conclusion matches the paper’s statement, the model’s proof is invalid because it rests on an incorrect exact identity rather than the asymptotic framework developed in the paper (see equation (1.8), definition (1.9), Theorem 0.1, and Remark 1.16).