Regular sequences and random walks in affine buildings
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 675-707.

We define and characterise regular sequences in affine buildings, thereby giving the p-adic analogue of the fundamental work of Kaimanovich on regular sequences in symmetric spaces. As applications we prove limit theorems for random walks on affine buildings and their automorphism groups.

On donne la définition et des caractérisations de suites régulières dans les immeubles affines. Ce faisant, on obtient l’analogue p-adique du travail fondamental de Kaimanovich sur les suites régulières dans les espaces symétriques. Comme application, nous démontrons des théorèmes limite pour des marches aléatoires dans les immeubles affines et leurs groupes d’automorphismes.

DOI: 10.5802/aif.2941
Classification: 20E42, 51E24, 05C81, 60J10
Keywords: Affine building, CAT(0), multiplicative ergodic theorem, random walks, regular sequences
Mot clés : Immeuble affine, CAT(0), théorème ergodique multiplicatif, marches aléatoires, suites régulières
Parkinson, James 1; Woess, Wolfgang 2

1 School of Mathematics and Statistics University of Sydney Carslaw Building, F07 NSW, 2006 (Australia)
2 Institut für Mathematische Strukturthorie Technische Universität Graz Steyrergasse 30 8010 Graz (Austria)
@article{AIF_2015__65_2_675_0,
     author = {Parkinson, James and Woess, Wolfgang},
     title = {Regular sequences and random walks in affine buildings},
     journal = {Annales de l'Institut Fourier},
     pages = {675--707},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {2},
     year = {2015},
     doi = {10.5802/aif.2941},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2941/}
}
TY  - JOUR
AU  - Parkinson, James
AU  - Woess, Wolfgang
TI  - Regular sequences and random walks in affine buildings
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 675
EP  - 707
VL  - 65
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2941/
DO  - 10.5802/aif.2941
LA  - en
ID  - AIF_2015__65_2_675_0
ER  - 
%0 Journal Article
%A Parkinson, James
%A Woess, Wolfgang
%T Regular sequences and random walks in affine buildings
%J Annales de l'Institut Fourier
%D 2015
%P 675-707
%V 65
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2941/
%R 10.5802/aif.2941
%G en
%F AIF_2015__65_2_675_0
Parkinson, James; Woess, Wolfgang. Regular sequences and random walks in affine buildings. Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 675-707. doi : 10.5802/aif.2941. http://www.numdam.org/articles/10.5802/aif.2941/

[1] Abramenko, Peter; Brown, Kenneth S. Buildings, Graduate Texts in Mathematics, 248, Springer, New York, 2008, pp. xxii+747 (Theory and applications) | DOI | MR | Zbl

[2] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, pp. xii+300 (Translated from the 1968 French original by Andrew Pressley) | DOI | MR | Zbl

[3] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999, pp. xxii+643 | DOI | MR | Zbl

[4] Brofferio, S. Renewal theory on the affine group of an oriented tree, J. Theoret. Probab., Volume 17 (2004) no. 4, pp. 819-859 | DOI | MR | Zbl

[5] Bruhat, F.; Tits, J. Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. (1972) no. 41, pp. 5-251 | DOI | Numdam | MR | Zbl

[6] Carter, Roger W. Simple groups of Lie type, John Wiley & Sons, London-New York-Sydney, 1972, pp. viii+331 (Pure and Applied Mathematics, Vol. 28) | MR | Zbl

[7] Cartwright, D. I.; Kaĭmanovich, V. A.; Woess, W. Random walks on the affine group of local fields and of homogeneous trees, Ann. Inst. Fourier (Grenoble), Volume 44 (1994) no. 4, pp. 1243-1288 | DOI | Numdam | MR | Zbl

[8] Cartwright, Donald I.; Woess, Wolfgang Isotropic random walks in a building of type à d , Math. Z., Volume 247 (2004) no. 1, pp. 101-135 | DOI | MR | Zbl

[9] Ji, Lizhen Buildings and their applications in geometry and topology, Asian J. Math., Volume 10 (2006) no. 1, pp. 11-80 | DOI | MR | Zbl

[10] Kaĭmanovich, V. A. Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 164 (1987) no. Differentsialnaya Geom. Gruppy Li i Mekh. IX, p. 29-46, 196–197 | DOI | MR | Zbl

[11] Karlsson, Anders; Ledrappier, François On laws of large numbers for random walks, Ann. Probab., Volume 34 (2006) no. 5, pp. 1693-1706 | DOI | MR | Zbl

[12] Karlsson, Anders; Margulis, Gregory A. A multiplicative ergodic theorem and nonpositively curved spaces, Comm. Math. Phys., Volume 208 (1999) no. 1, pp. 107-123 | DOI | MR | Zbl

[13] Lindlbauer, Marc; Voit, Michael Limit theorems for isotropic random walks on triangle buildings, J. Aust. Math. Soc., Volume 73 (2002) no. 3, pp. 301-333 | DOI | MR | Zbl

[14] Macdonald, I. G. Spherical functions on a group of p -adic type, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, 1971, pp. vii+79 (Publications of the Ramanujan Institute, No. 2) | MR | Zbl

[15] Oseledec, V. I. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., Volume 19 (1968), pp. 179-210 | MR | Zbl

[16] Parkinson, James Buildings and Hecke algebras, J. Algebra, Volume 297 (2006) no. 1, pp. 1-49 | DOI | MR | Zbl

[17] Parkinson, James Spherical harmonic analysis on affine buildings, Math. Z., Volume 253 (2006) no. 3, pp. 571-606 | DOI | MR | Zbl

[18] Parkinson, James Isotropic random walks on affine buildings, Ann. Inst. Fourier (Grenoble), Volume 57 (2007) no. 2, pp. 379-419 | DOI | Numdam | MR | Zbl

[19] Parkinson, James; Schapira, Bruno A local limit theorem for random walks on the chambers of à 2 buildings, Random walks, boundaries and spectra (Progr. Probab.), Volume 64, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 15-53 | DOI | MR | Zbl

[20] Ronan, M. A. A construction of buildings with no rank 3 residues of spherical type, Buildings and the geometry of diagrams (Como, 1984) (Lecture Notes in Math.), Volume 1181, Springer, Berlin, 1986, pp. 242-248 | DOI | MR | Zbl

[21] Ronan, Mark Lectures on buildings, University of Chicago Press, Chicago, IL, 2009, pp. xiv+228 (Updated and revised) | MR | Zbl

[22] Sawyer, Stanley Isotropic random walks in a tree, Z. Wahrsch. Verw. Gebiete, Volume 42 (1978) no. 4, pp. 279-292 | DOI | MR | Zbl

[23] Schapira, Bruno Random walk on a building of type à r and Brownian motion of the Weyl chamber, Ann. Inst. Henri Poincaré Probab. Stat., Volume 45 (2009) no. 2, pp. 289-301 | DOI | Numdam | MR | Zbl

[24] Steinberg, Robert Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968, pp. iii+277 (Notes prepared by John Faulkner and Robert Wilson) | MR | Zbl

[25] Tits, Jacques Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974, pp. x+299 | MR | Zbl

[26] Tits, Jacques Immeubles de type affine, Buildings and the geometry of diagrams (Como, 1984) (Lecture Notes in Math.), Volume 1181, Springer, Berlin, 1986, pp. 159-190 | DOI | MR | Zbl

[27] Tolli, Filippo A local limit theorem on certain p-adic groups and buildings, Monatsh. Math., Volume 133 (2001) no. 2, pp. 163-173 | DOI | MR | Zbl

[28] Weiss, Richard M. The structure of affine buildings, Annals of Mathematics Studies, 168, Princeton University Press, Princeton, NJ, 2009, pp. xii+368 | MR | Zbl

Cited by Sources: