P. H. Bérard, Spectres et groupes cristallographiques I: Domaines euclidiens, Inventiones Mathematicae, vol.1, issue.2, pp.179-199, 1980.
DOI : 10.1007/BF01403168

P. Biane, MINUSCULE WEIGHTS AND RANDOM WALKS ON LATTICES, Quant. Prob. Rel. Topics, vol.7, pp.51-65, 1992.
DOI : 10.1142/9789814354783_0004

. Ph, P. Biane, N. O. Bougerol, and . Connell, Littelmann paths and Brownian paths, Duke Math, J, vol.130, issue.1, pp.127-167, 2005.

A. Borodin and V. Gorin, Lectures on integrable probability, arXiv preprint arXiv:1212, p.2012

T. Bröcker and T. Dieck, Representations of compact Lie groups, Graduate texts in Mathematics, vol.98, 1985.
DOI : 10.1007/978-3-662-12918-0

A. H. Dooley and N. J. Wildberger, Harmonic analysis and the global exponential map for compact Lie groups, Functional Analysis and Its Applications, vol.29, issue.3, pp.2532-1225907, 1993.
DOI : 10.1007/BF01768664

S. N. Ethier and T. G. Kurtz, Markov Processes : Characterization and Convergence, Math. Stat, 1986.
DOI : 10.1002/9780470316658

J. Fuchs, Affine Lie Algebras and Quantum Groups, 1992.

I. M. Gessel and D. Zeilberger, Random walk in a Weyl chamber, Proceedings of the American Mathematical Society, vol.115, issue.1, pp.27-31, 1992.
DOI : 10.1090/S0002-9939-1992-1092920-8

D. J. Grabiner, Random Walk in an Alcove of an Affine Weyl Group, and Non-colliding Random Walks on an Interval, Journal of Combinatorial Theory, Series A, vol.97, issue.2, pp.285-306, 2002.
DOI : 10.1006/jcta.2001.3216

J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol.9, 1978.
DOI : 10.1007/978-1-4612-6398-2

V. G. Kac, Infinite dimensional Lie algebras, 1990.

A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, vol.64, 2004.
DOI : 10.1090/gsm/064

C. Krattenthaler, Asymptotics For Random Walks In Alcoves Of Affine Weyl Groups, Journal of Combinatorial Theory Series A, vol.97, pp.285-306, 2003.

C. Lecouvey, E. Lesigne, and M. Peigné, Conditioned random walks from Kac-Moody root systems, Transactions of the American Mathematical Society, vol.368, issue.5, p.2013
DOI : 10.1090/tran/6468
URL : https://hal.archives-ouvertes.fr/hal-00833657

P. Littelmann, Paths and Root Operators in Representation Theory, The Annals of Mathematics, vol.142, issue.3, pp.499-525, 1995.
DOI : 10.2307/2118553

J. F. Van-diejen and E. Emsiz, Discrete harmonic analysis on a Weyl alcove, Journal of Functional Analysis, vol.265, issue.9, 2012.
DOI : 10.1016/j.jfa.2013.06.023

M. Walton, Affine Kac-Moody Algebras and the Wess-Zumino-Witten Model, pp.hep- th, 1999.

D. Wehn, Limit distributions on Lie groups, 1960.

N. J. Wildberger, Finite commutative hypergroups and applications from group theory to conformal field theory, Applications of Hypergroups and Related Measure Algebras, Contemp. Math, vol.183, 1995.

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