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, 2014, 34 (11) : 4689-4717. doi: 10.3934/dcds.2014.34.4689
References:
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show all references

References:
[1]

Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 337-348. doi: 10.3934/dcdss.2009.2.337.  Google Scholar

[2]

Monatsh. Math., 167 (2012), 333-356. doi: 10.1007/s00605-012-0376-y.  Google Scholar

[3]

H. Aytac, J. M. Freitas and S. Vaienti, Laws of rare events for deterministic and random dynamical systems,, To appear in Trans. Amer Math. Soc., ().   Google Scholar

[4]

B. Bekka and Y. Guivarc'h, On the spectral theory of groups of affine transformations of compact nilmanifolds,, , ().   Google Scholar

[5]

Ann. of Math., 178 (2013), 1017-1059. doi: 10.4007/annals.2013.178.3.5.  Google Scholar

[6]

(French) [Stationary measures and invariant subsets of homogeneous spaces], Ann. of Math., 174 (2011), 1111-1162. doi: 10.4007/annals.2011.174.2.8.  Google Scholar

[7]

J. Amer. Math. Soc., 26 (2013), 659-734. doi: 10.1090/S0894-0347-2013-00760-2.  Google Scholar

[8]

Invent. Math., 187 (2012), 37-59. doi: 10.1007/s00222-011-0328-5.  Google Scholar

[9]

J. Amer. Math. Soc., 24 (2011), 231-280. doi: 10.1090/S0894-0347-2010-00674-1.  Google Scholar

[10]

Ergodic Theory Dynam. Systems, 21 (2001), 401-420. doi: 10.1017/S0143385701001201.  Google Scholar

[11]

Walter de Gruyter GmbH & Co. KG,, Berlin, Random walks and geometry (2004), 431-444.  Google Scholar

[12]

Ergodic Theory Dynam. Systems, 28 (2008), 1117-1133. doi: 10.1017/S0143385707000624.  Google Scholar

[13]

MR2639719, 147 (2010), 675-710. doi: 10.1007/s00440-009-0221-y.  Google Scholar

[14]

J. Stat. Phys., 142 (2011), 108-126. doi: 10.1007/s10955-010-0096-4.  Google Scholar

[15]

Adv. Math., 231 (2012), 2626-2665. doi: 10.1016/j.aim.2012.07.029.  Google Scholar

[16]

C. R. Math. Acad. Sci. Paris, 351 (2013), 703-705. doi: 10.1016/j.crma.2013.09.017.  Google Scholar

[17]

Ergodic Theory Dynam. Systems, 30 (2010), 757-771. doi: 10.1017/S0143385709000406.  Google Scholar

[18]

Ergodic Theory Dynam. Systems, 31 (2011), 1363-1390. doi: 10.1017/S014338571000057X.  Google Scholar

[19]

Trans. Amer. Math. Soc., 364 (2012), 661-688. doi: 10.1090/S0002-9947-2011-05271-2.  Google Scholar

[20]

Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.  Google Scholar

[21]

Spectral analysis in geometry and number theory, Contemp. Math., 484 (2009), 177-185. doi: 10.1090/conm/484/09474.  Google Scholar

[22]

Springer-Verlag, New York-Berlin, 1983.  Google Scholar

[23]

Ann. Inst. Fourier (Grenoble), 50 (2000), 833-863. doi: 10.5802/aif.1775.  Google Scholar

[24]

Acta Math., 149 (1982), 215-237. doi: 10.1007/BF02392354.  Google Scholar

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