Noyau de la chaleur et discretisation d’une variete riemannienne

One considers a bounded geometry non-compact Riemannian manifold, and the graph obtained by discretizing this manifold. One shows that the uniform decay for large time of the heat kernel on the manifold and the decay of the standard random walk on the graph are the same, in the polynomial scale. As a consequence, such a large time behaviour of the heat kernel is invariant under rough isometries.