# American Institute of Mathematical Sciences

November  2014, 34(11): 4689-4717. doi: 10.3934/dcds.2014.34.4689

## Extreme value theory for random walks on homogeneous spaces

 1 School of Mathematics, University of Bristol, Bristol, United Kingdom

Received  April 2013 Revised  February 2014 Published  May 2014

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.
Citation: Maxim Sølund Kirsebom. Extreme value theory for random walks on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4689-4717. doi: 10.3934/dcds.2014.34.4689
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