Local limit theorem of Brownian motion on metric trees

Let T be a locally finite tree whose geometric boundary has infinitely many points. Suppose that a non-amenable group acts isometrically and geometrically on the tree T. In this paper, we show that if the length spectrum is Diophantine, then there exists a continuous function C on T² such that the heat kernel p (t, x, y) of T satisfies ₓ t^3/2e^₀tp (t, x, y) =C (x, y) for any x, y T. Here, ₀ is the bottom of the spectrum of the Laplacian on T.