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Spectral gap properties for random walks on homogeneous spaces: examples and consequences

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Interactions between Dynamics of Group Actions and Number Theory

Co-authors: J.-P Conze, B. Bekka, E .LePage

Let E be a homogeneous space of a Lie group G, p a finitely supported probability measure on G, such that supp(p) generates topologically G. We show that, in various situations, convolution by p has a spectral gap on some suitable functional space on E . We consider in particular Hilbert spaces and Holder spaces on E and actions by affine transformations. If G is the motion group of Euclidean space V ,we get equidistribution of the random walk on V. If G is the affine group of V,p has a stationary probability, and the projection of p on GL(V) satisfies "generic" conditions we get that the random walk satisfies Frechet's extreme law , and Sullivan's Logarithm law.

This talk is part of the Isaac Newton Institute Seminar Series series.

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