Heat kernel and Green function estimates on affine buildings of type ${\tilde{A}}_r$

Abstract

We obtain a global estimate of the transition density $p^n(0,x)$ associated to a nearest neighbor random walk, called here "simple", on affine buildings of type ${\tilde{A}}_r$. Then we deduce a global estimate of the Green function. This is the analogue of a result on Riemannian symmetric spaces of the noncompact type.