Rate of decay of concentration functions on discrete groups
Given a probability measure \(\mu\) on a locally compact group \(G\) one defines the concentration functions to be \(f_n(K)= \sup_{g\in G}\,\mu^{(n)} (Kg)\) where \(K\) is a compact subset of \(g\) and \(\mu^{(n)}\) is the \(n\)th convolution power of \(\nu\). The work of \textit{K. H. Hofmann} and \textit{A. Mukherjea} [Math. Ann. 256, 535--548 (1981; Zbl 0471.60015)] was completed by \textit{W. Jaworski}, \textit{J. Rosenblatt} and \textit{G. Willis} [ibid. 305, 673--691 (1996; Zbl 0854.43001)] to show that if \(G\) is non-compact and \(\mu\) is irreducible, then \(\lim_{n\to\infty}\, f_n(K)= 0\) for any compact set \(K\). The rate of this convergence has been studied by a variety of authors, including \textit{P. Bougerol} [C. R. Acad. Sci., Paris, Sér. A 283, 527--529 (1976; Zbl 0368.60010) and in: Probability measures on groups. Lect. Notes Math. 706, 36--40 (1979; Zbl 0404.60019)], \textit{N. Th. Varopoulos} [in: Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 951--957 (1991; Zbl 0744.43006)], and \textit{N. Th. Varopoulos}, \textit{L. Saloff-Coste} and \textit{T. Coulhon} [``Analysis on Lie groups'' (1992; Zbl 0813.22003)]. In this note the rate of this convergence is presented in the case where \(G\) is a non-locally finite discrete group. In particular it is shown that if the volume growth \(V(m)\) of \(G\) satisfies \(V(m)\geq cm^D\), then for any compact set \(K\) it follows \(f_n(K)\leq Cn^{-D/2}\) (Theorem 1 -- Main theorem). The references contain 7 bibliographical hints.
scientific article; zbMATH DE number 1960100