A local limit theorem on certain \(p\)-adic groups and buildings (Q1601791)

The author investigates local central limit theorems for finite range random walks on certain \(p\)-adic groups and buildings. In particular, for biinvariant, aperiodic irreducible random walks with finite range, an explicit formula is derived together with Berry-Esseen-type estimates for the error. This result is closely related to the local limit theorems on the \(\widetilde A_d\)-buildings in recent papers of \textit{D. Cartwright} and \textit{W. Woess} (Preprint) and \textit{M. Lindlbauer} and the reviewer [J. Aust. Math. Soc. 73, 301-334 (2002)]. Moreover, the author proves a local limit theorem for arbitrary aperiodic, symmetric, and irreducible random walks with finite range on certain \(p\)-adic groups. This interesting result supplements well-known results of S. Lalley and others for not necessarily isotropic random walks on free groups and homogeneous trees.