Abstract
In this paper we study extreme events for random walks on homogeneous spaces. We consider the following three cases. On the torus we study closest returns of a random walk to a fixed point in the space. For a random walk on the space of unimodular lattices we study extreme values for lengths of the shortest vector in a lattice. For a random walk on a homogeneous space we study the maximal distance a random walk gets away from an arbitrary fixed point in the space. We prove an exact limiting distribution on the torus and upper and lower bounds for sparse subsequences of random walks in the two other cases. In all three settings we obtain a logarithm law.
Highlights
Let X be a probability space and G a group acting on X
K > 0 is again the constant from Remark 1.7. This is a random walk analogue of the logarithm law Kleinbock and Margulis proved for geodesics
A natural question to ask is whether one could determine the extreme value distribution for the geodesic flow, since it would be a generalization of the logarithm law mentioned
Summary
Let X be a probability space and G a group acting on X. We are interested in the closest returns of a random walk to a fixed point on the torus and in particular, how these shortest distances distribute. K > 0 is again the constant from Remark 1.7 This is a random walk analogue of the logarithm law Kleinbock and Margulis proved for geodesics. A natural question to ask is whether one could determine the extreme value distribution for the geodesic flow, since it would be a generalization of the logarithm law mentioned One result in this direction is by Pollicott [21]. For the case of the torus an additional argument is required which we give as well
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