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Lower bounds for random walks on amenable \(p\)-adic groups. (Bornes inférieures pour les marches aléatoires sur les groupes \(p\)-adiques moyennables.) (French) Zbl 1102.60041

Let \(G\) be a locally compact, amenable group. Denote by \(e\) the identity of \(G\) and by \(d^rg\) the right Haar measure. Suppose that \(d\mu(g)=\varphi(g)d^rg\) is a symmetric probability measure on \(G\), with \(\varphi(g)\) a function with compact support. Denote by \(d\mu^{*n}(g)=d(\mu*\mu*\dotsb*\mu)(g)=\varphi_n(g)d^rg\) the \(n\)th convolution power of \(d\mu(g)\). One of the central problems in the study of the random walk described by the convolution powers of \(d\mu(g)\), is to compute the rate of convergence in the limit \(\lim_{n\to\infty}\varphi(e)=0\). For instance, if \(G\) is an amenable, unimodular Lie group, one has the dicotomy \[ \varphi_n(e)\approx n^{-D/2}\Leftrightarrow \gamma(n)\approx n^D,\quad \varphi_n(e)\approx \exp(-n^{1/3})\Leftrightarrow \gamma(n)\approx \exp(n), \] where \(\gamma(n)\) is the volume growth of \(G\) [see N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, “Analysis on Lie groups” (1992; Zbl 0813.22003)]. In a more general situation, where \(G\) is only a locally compact, compactly generated group of exponential growth, one has only the upper estimate \[ \varphi_n(e)\leq C\exp(-cn^{1/3}),\quad n\leq 1; \] see W. Hebisch and L. Saloff-Coste [Ann. Probab. 21, No. 2, 673–709 (1993; Zbl 0776.60086)]. Now let \(k\) be a totally disconnected, non-discrete, locally compact field of characteristic zero, and let \(G\) a connected, algebraic group over \(k\). The paper examines the case in which \(G\) is also compactly generated and amenable. Under the hypothesis that \(\varphi\) is continuous and there exists a symmetric subset \(\Omega\subset G\), containing \(e\) and such that
\[ \inf\{\varphi(g):g\in \Omega\}>0,\quad G=\bigcup_{n\leq 0}\Omega^n, \] the author proves the following lower estimate: \[ \varphi_{2n}(e)\geq c\exp(-Cn^{1/3}),\quad n\geq 1. \] Under the additional hypothesis that \(G\) is unimodular, there is also a lower estimate: \[ \frac{1}{C}\exp(-c_1n^{1/3})\leq\varphi_{2n}(e)\leq C\exp(-c_2n^{1/3}). \]

MSC:

60G50 Sums of independent random variables; random walks
22E35 Analysis on \(p\)-adic Lie groups
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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