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
References:
[1]

J. S. Athreya, A. Ghosh and A. Prasad, Ultrametric logarithm laws. I,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 337.  doi: 10.3934/dcdss.2009.2.337.  Google Scholar

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J. S. Athreya, A. Ghosh and A. Prasad, Ultrametric logarithm laws, II,, Monatsh. Math., 167 (2012), 333.  doi: 10.1007/s00605-012-0376-y.  Google Scholar

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

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Y. Benoist and J. F. Quint, Stationary measures and invariant subsets of homogeneous spaces (III),, Ann. of Math., 178 (2013), 1017.  doi: 10.4007/annals.2013.178.3.5.  Google Scholar

[6]

Y. Benoist and J. F. Quint, Mesures stationnaires et fermés invariants des espaces homogenes,, (French) [Stationary measures and invariant subsets of homogeneous spaces], 174 (2011), 1111.  doi: 10.4007/annals.2011.174.2.8.  Google Scholar

[7]

Y. Benoist and J. F. Quint, Stationary measures and invariant subsets of homogeneous spaces (II),, J. Amer. Math. Soc., 26 (2013), 659.  doi: 10.1090/S0894-0347-2013-00760-2.  Google Scholar

[8]

Y. Benoist and J. F. Quint, Random walks on finite volume homogeneous spaces,, Invent. Math., 187 (2012), 37.  doi: 10.1007/s00222-011-0328-5.  Google Scholar

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J. Bourgain, A. Furman, E. Lindenstrauss and S. Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus,, J. Amer. Math. Soc., 24 (2011), 231.  doi: 10.1090/S0894-0347-2010-00674-1.  Google Scholar

[10]

P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems,, Ergodic Theory Dynam. Systems, 21 (2001), 401.  doi: 10.1017/S0143385701001201.  Google Scholar

[11]

A. Eskin and G. A. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds,, Walter de Gruyter GmbH & Co. KG, (2004), 431.   Google Scholar

[12]

A. C. M. Freitas and J. M. Freitas, Extreme values for Benedicks-Carleson quadratic maps,, Ergodic Theory Dynam. Systems, 28 (2008), 1117.  doi: 10.1017/S0143385707000624.  Google Scholar

[13]

A. C. M. Freitas, J. M. Freitas and M. Todd, Hitting time statistics and extreme value theory,, MR2639719, 147 (2010), 675.  doi: 10.1007/s00440-009-0221-y.  Google Scholar

[14]

A. C. M. Freitas, J. M. Freitas and M. Todd, Extreme value laws in dynamical systems for non-smooth observations,, J. Stat. Phys., 142 (2011), 108.  doi: 10.1007/s10955-010-0096-4.  Google Scholar

[15]

A. C. M. Freitas, J. M. Freitas and M. Todd, The extremal index, hitting time statistics and periodicity,, Adv. Math., 231 (2012), 2626.  doi: 10.1016/j.aim.2012.07.029.  Google Scholar

[16]

Y. Guivarc'h and E. Le Page, Extreme-value asymptotics for affine random walks,, C. R. Math. Acad. Sci. Paris, 351 (2013), 703.  doi: 10.1016/j.crma.2013.09.017.  Google Scholar

[17]

C. Gupta, Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems,, Ergodic Theory Dynam. Systems, 30 (2010), 757.  doi: 10.1017/S0143385709000406.  Google Scholar

[18]

C. Gupta, M. Holland and M. Nicol, Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps,, Ergodic Theory Dynam. Systems, 31 (2011), 1363.  doi: 10.1017/S014338571000057X.  Google Scholar

[19]

M. Holland, M. Nicol and A. Török, Extreme value theory for non-uniformly expanding dynamical systems,, Trans. Amer. Math. Soc., 364 (2012), 661.  doi: 10.1090/S0002-9947-2011-05271-2.  Google Scholar

[20]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces,, Invent. Math., 138 (1999), 451.  doi: 10.1007/s002220050350.  Google Scholar

[21]

M. Pollicott, Limiting distributions for geodesics excursions on the modular surface,, Spectral analysis in geometry and number theory, 484 (2009), 177.  doi: 10.1090/conm/484/09474.  Google Scholar

[22]

H. Rootzen, M. R. Leadbetter and G. Lindgren, Extremes and Related Properties of Random Sequences and Processes,, Springer-Verlag, (1983).   Google Scholar

[23]

Y. Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups,, Ann. Inst. Fourier (Grenoble), 50 (2000), 833.  doi: 10.5802/aif.1775.  Google Scholar

[24]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics,, Acta Math., 149 (1982), 215.  doi: 10.1007/BF02392354.  Google Scholar

show all references

References:
[1]

J. S. Athreya, A. Ghosh and A. Prasad, Ultrametric logarithm laws. I,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 337.  doi: 10.3934/dcdss.2009.2.337.  Google Scholar

[2]

J. S. Athreya, A. Ghosh and A. Prasad, Ultrametric logarithm laws, II,, Monatsh. Math., 167 (2012), 333.  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]

Y. Benoist and J. F. Quint, Stationary measures and invariant subsets of homogeneous spaces (III),, Ann. of Math., 178 (2013), 1017.  doi: 10.4007/annals.2013.178.3.5.  Google Scholar

[6]

Y. Benoist and J. F. Quint, Mesures stationnaires et fermés invariants des espaces homogenes,, (French) [Stationary measures and invariant subsets of homogeneous spaces], 174 (2011), 1111.  doi: 10.4007/annals.2011.174.2.8.  Google Scholar

[7]

Y. Benoist and J. F. Quint, Stationary measures and invariant subsets of homogeneous spaces (II),, J. Amer. Math. Soc., 26 (2013), 659.  doi: 10.1090/S0894-0347-2013-00760-2.  Google Scholar

[8]

Y. Benoist and J. F. Quint, Random walks on finite volume homogeneous spaces,, Invent. Math., 187 (2012), 37.  doi: 10.1007/s00222-011-0328-5.  Google Scholar

[9]

J. Bourgain, A. Furman, E. Lindenstrauss and S. Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus,, J. Amer. Math. Soc., 24 (2011), 231.  doi: 10.1090/S0894-0347-2010-00674-1.  Google Scholar

[10]

P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems,, Ergodic Theory Dynam. Systems, 21 (2001), 401.  doi: 10.1017/S0143385701001201.  Google Scholar

[11]

A. Eskin and G. A. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds,, Walter de Gruyter GmbH & Co. KG, (2004), 431.   Google Scholar

[12]

A. C. M. Freitas and J. M. Freitas, Extreme values for Benedicks-Carleson quadratic maps,, Ergodic Theory Dynam. Systems, 28 (2008), 1117.  doi: 10.1017/S0143385707000624.  Google Scholar

[13]

A. C. M. Freitas, J. M. Freitas and M. Todd, Hitting time statistics and extreme value theory,, MR2639719, 147 (2010), 675.  doi: 10.1007/s00440-009-0221-y.  Google Scholar

[14]

A. C. M. Freitas, J. M. Freitas and M. Todd, Extreme value laws in dynamical systems for non-smooth observations,, J. Stat. Phys., 142 (2011), 108.  doi: 10.1007/s10955-010-0096-4.  Google Scholar

[15]

A. C. M. Freitas, J. M. Freitas and M. Todd, The extremal index, hitting time statistics and periodicity,, Adv. Math., 231 (2012), 2626.  doi: 10.1016/j.aim.2012.07.029.  Google Scholar

[16]

Y. Guivarc'h and E. Le Page, Extreme-value asymptotics for affine random walks,, C. R. Math. Acad. Sci. Paris, 351 (2013), 703.  doi: 10.1016/j.crma.2013.09.017.  Google Scholar

[17]

C. Gupta, Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems,, Ergodic Theory Dynam. Systems, 30 (2010), 757.  doi: 10.1017/S0143385709000406.  Google Scholar

[18]

C. Gupta, M. Holland and M. Nicol, Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps,, Ergodic Theory Dynam. Systems, 31 (2011), 1363.  doi: 10.1017/S014338571000057X.  Google Scholar

[19]

M. Holland, M. Nicol and A. Török, Extreme value theory for non-uniformly expanding dynamical systems,, Trans. Amer. Math. Soc., 364 (2012), 661.  doi: 10.1090/S0002-9947-2011-05271-2.  Google Scholar

[20]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces,, Invent. Math., 138 (1999), 451.  doi: 10.1007/s002220050350.  Google Scholar

[21]

M. Pollicott, Limiting distributions for geodesics excursions on the modular surface,, Spectral analysis in geometry and number theory, 484 (2009), 177.  doi: 10.1090/conm/484/09474.  Google Scholar

[22]

H. Rootzen, M. R. Leadbetter and G. Lindgren, Extremes and Related Properties of Random Sequences and Processes,, Springer-Verlag, (1983).   Google Scholar

[23]

Y. Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups,, Ann. Inst. Fourier (Grenoble), 50 (2000), 833.  doi: 10.5802/aif.1775.  Google Scholar

[24]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics,, Acta Math., 149 (1982), 215.  doi: 10.1007/BF02392354.  Google Scholar

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