LOCAL LIMIT THEOREM OF BROWNIAN MOTION ON METRIC TREES

The paper proves a t^{-3/2}e^{-λ0 t} local limit for the heat kernel on a locally finite metric tree under a geometric Γ-action with a dense, Diophantine length spectrum, via a Gibbs/thermodynamic formalism for a potential f_λ on the geodesic flow, coding by a finite Markov shift, and a uniform rapid-mixing statement; the key singular behavior is captured by the limit formula lim_{λ→λ0−} (λ0−λ)^{-1/2} ∂_λ G_λ(x,y) = c(x,y) and a Tauberian step (Theorem 1.2 and display (1.2)) . The model’s outline asserts a different operator-theoretic proof based on an explicit Ruelle transfer-operator representation of the resolvent and a Dolgopyat-type spectral gap, but it contains critical inaccuracies: (i) the sign/half-plane for the Laplace transform is reversed relative to the paper (the paper defines G_λ with e^{λ t} and studies λ→λ0−, not λ→λ0^+), (ii) it assumes, without justification here, a concrete Neumann-series/renewal identification of G_λ(x,y) as a matrix element of (I−L_λ)^{-1}, and (iii) it postulates a uniform spectral gap for twisted operators from the Diophantine condition without proving the necessary non-integrability/transversality hypotheses. While the model’s conclusion matches the paper’s theorem, its argument is not correct as written, and key steps are unproven or misstated.