Extreme value theory for random walks on homogeneous spaces

Maxim Sølund Kirsebom

doi:10.3934/dcds.2014.34.4689

Article Contents

2014, Volume 34, Issue 11: 4689-4717. Doi: 10.3934/dcds.2014.34.4689

This issue Previous Article Supercritical problems in domains with thin toroidal holes Next Article On some Liouville type theorems for the compressible Navier-Stokes equations

Extreme value theory for random walks on homogeneous spaces

Maxim Sølund Kirsebom^1,

1.
School of Mathematics, University of Bristol, Bristol

Received: April 2013

Revised: February 2014

Published: November 2014

Abstract / Introduction Related Papers Cited by

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.

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Mathematics Subject Classification: Primary: 82C41, 60G70; Secondary: 43A85.

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