Random walks in Dirichlet environment: an overview
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 463-509.

Les marches aléatoires en environnement de Dirichlet (RWDE) correspondent à des marches alétoires en environnement aléatoire dont les probabilités de transition en chaque site sont indépendantes et distribuées suivant une même loi de Dirichlet. Le modèle est donc paramétré par une famille de poids (α i ) i=1,...,2d , un pour chaque direction dirigée de d . Dans ce cas, la loi moyennée est celle de la marche renforcée avec renforcement linéaire sur les arêtes orientées. Les RWDE ont une propriété remarquable d’invariance en loi par retournement du temps, de laquelle découlent plusieurs résultats encore inaccessibles dans le cas général, comme la propriété d’équivalence des points de vue statiques et dynamiques ou comme la caractérisation des régimes de transience directionnelle et de ballisticité. Dans cet article, nous présentons les développements récents sur ce modèle et donnons plusieurs esquisses de démonstrations mettant en relief les arguments centraux du sujet. Nous présentons aussi des calculs nouveaux sur la fonction de taux des grandes déviations dans le cas unidimensionnel.

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on d where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by a family of positive weights (α i ) i=1,...,2d , one for each oriented direction of d . In this case, the annealed law is that of a reinforced random walk, with linear reinforcement on directed edges. RWDE have a remarkable property of statistical invariance by time reversal from which can be inferred several properties that are still inaccessible for general environments, such as the equivalence of static and dynamic points of view and a description of the directionally transient and ballistic regimes. In this paper we review the recent developments on this model and give several sketches of proofs presenting the core of the arguments. We also present new computations of the large deviation rate function for one dimensional RWDE.

Publié le :
DOI : https://doi.org/10.5802/afst.1542
Classification : 60K37,  60K35
Mots clés : Random walk in random environment, Dirichlet distribution, Reinforced random walks, invariant measure viewed from the particle
@article{AFST_2017_6_26_2_463_0,
     author = {Sabot, Christophe and Tournier, Laurent},
     title = {Random walks in Dirichlet environment: an overview},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {463--509},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {2},
     year = {2017},
     doi = {10.5802/afst.1542},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1542/}
}
Sabot, Christophe; Tournier, Laurent. Random walks in Dirichlet environment: an overview. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 463-509. doi : 10.5802/afst.1542. http://www.numdam.org/articles/10.5802/afst.1542/

[1] Aharoni, Ron; Berger, Eli; Georgakopoulos, Agelos; Perlstein, Amitai; Sprüssel, Philipp The max-flow min-cut theorem for countable networks, J. Comb. Theory, Volume 101 (2011) no. 1, pp. 1-17 | Article

[2] Angel, Omer; Crawford, Nicholas; Kozma, Gady Localization for linearly edge reinforced random walks, Duke Math. J., Volume 163 (2014) no. 5, pp. 889-921 | Article

[3] Barraquand, Guillaume; Corwin, Ivan Random-walk in Beta-distributed random environment (2015) (https://arxiv.org/abs/1503.04117)

[4] Berger, Noam; Cohen, Moran; Rosenthal, Ron Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d environments (2014) (https://arxiv.org/abs/1405.6819v1)

[5] Berger, Noam; Drewitz, Alexander; Ramírez, Alejandro F. Effective polynomial ballisticity conditions for random walk in random environment, Comm. Pure Appl. Math., Volume 67 (2014) no. 12, pp. 1947-1973 | Article

[6] Berger, Noam; Zeitouni, Ofer A quenched invariance principle for certain ballistic random walks in i.i.d. environments, In and Out of Equilibrium 2 (Progress in Probability), Volume 60, Birkhäuser, 2008, pp. 137-160

[7] Bolthausen, Erwin; Sznitman, Alain-Sol Ten lectures on random media, DMV Seminar, 32, Birkhäuser, 2002, vi+116 pages | Article

[8] Bolthausen, Erwin; Zeitouni, Ofer Multiscale analysis of exit distributions for random walks in random environments, Probab. Theory Relat. Fields, Volume 138 (2007) no. 3–4, pp. 581-645 | Article

[9] Bouchet, Élodie Sub-ballistic random walk in Dirichlet environment, Electron. J. Probab., Volume 18 (2013) no. 58, pp. 1-25

[10] Bouchet, Élodie; Ramírez, Alejandro F.; Sabot, Christophe Sharp ellipticity conditions for ballistic behavior of random walks in random environment (2013) (https://arxiv.org/abs/1310.6281v1)

[11] Bouchet, Élodie; Sabot, Christophe; Dos Santos, Renato Soares A Quenched Functional Central Limit Theorem for Random Walks in Random Environments under (T) γ (2014) (https://arxiv.org/abs/1409.5528)

[12] Campos, David; Ramírez, Alejandro F. Ellipticity criteria for ballistic behavior of random walks in random environment, Probab. Theory Relat. Fields, Volume 160 (2014) no. 1–2, pp. 189-251 | Article

[13] Chamayou, Jean-François; Letac, Gérard Explicit stationary distributions for compositions of random functions and products of random matrices, J. Theor. Probab., Volume 4 (1991) no. 1, pp. 3-36 | Article

[14] Comets, Francis; Gantert, Nina; Zeitouni, Ofer Quenched, annealed and functional large deviations for one-dimensional random walk in random environment, Probab. Theory Relat. Fields, Volume 118 (2000) no. 1, pp. 65-114 (erratum ibid. 125, no. 1, p. 42-44) | Article

[15] Coppersmith, D.; Diaconis, P. Random walks with reinforcement (1998) (Unpublished manuscript)

[16] Durrett, Richard Probability: theory and examples, Duxbury advanced series, Thompson Brooks/Cole, 2005, xi+497 pages

[17] Enriquez, Nathanaël; Sabot, Christophe Random walks in a Dirichlet environment, Electron. J. Probab., Volume 11 (2006), pp. 802-817 (paper no. 31, electronic only) | Article

[18] Enriquez, Nathanaël; Sabot, Christophe; Tournier, Laurent; Zindy, Olivier Stable fluctuations for ballistic random walks in random environment on (2010) (https://arxiv.org/abs/1004.1333)

[19] Enriquez, Nathanaël; Sabot, Christophe; Zindy, Olivier Limit laws for transient random walks in random environment on , Ann. Inst. Fourier, Volume 59 (2009), pp. 2469-2508 | Article

[20] Enriquez, Nathanaël; Sabot, Christophe; Zindy, Olivier A probabilistic representation of constants in Kesten’s renewal theorem, Probab. Theory Relat. Fields, Volume 144 (2009) no. 3–4, pp. 581-613 | Article

[21] Ford, Lester R. jun.; Fulkerson, Delbert Ray Flows in networks, Princeton University Press, 1962, xii+194 pages

[22] Fribergh, Alexander; Kious, Daniel Local trapping for elliptic random walks in random environments in d (2014) (https://arxiv.org/abs/1404.2060v1)

[23] Greven, Andreas; den Hollander, Frank Large deviations for a random walk in random environment, Ann. Probab., Volume 22 (1994) no. 3, pp. 1381-1428 | Article

[24] den Hollander, Frank Large deviations, Fields Institute Monographs, 14, American Mathematical Society, 2000, x+143 pages

[25] Kalikow, Steven A. Generalized random walk in a random environment, Ann. Probab., Volume 9 (1981) no. 5, pp. 753-768 | Article

[26] Keane, Michael S.; Rolles, Silke W. W. Edge-reinforced random walk on finite graphs, Infinite dimensional stochastic analysis (Amsterdam, 1999), 217–234, R (K. Ned. Akad. Wet.), Volume 52 (2000), pp. 217-234

[27] Kesten, Harry Random difference equations and renewal theory for products of random matrices, Acta Mathematica, Volume 131 (1973) no. 1, pp. 207-248 | Article

[28] Kesten, Harry; Kozlov, Mykyta V.; Spitzer, Frank A limit law for random walk in a random environment, Compos. Math., Volume 30 (1975) no. 2, pp. 145-168

[29] Lawler, Gregory F. Weak convergence of a random walk in a random environment, Comm. Math. Phys., Volume 87 (1982/83) no. 1, pp. 81-87 http://projecteuclid.org/euclid.cmp/1103921905 | Article

[30] Levin, David A.; Peres, Yuval Pólya’s Theorem on Random Walks via Pólya’s Urn, Am. Math. Mon., Volume 117 (2010) no. 3, pp. 220-231 | Article

[31] Lyons, Russel; Peres, Yuval Probability on Trees and Networks, Cambridge University Press, 2015 (In preparation. Current version available at http://pages.iu.edu/~rdlyons/)

[32] Merkl, Franz; Rolles, Silke W. W. Recurrence of edge-reinforced random walk on a two-dimensional graph, Ann. Probab., Volume 37 (2009) no. 5, pp. 1679-1714 | Article

[33] Pemantle, Robin Phase transition in reinforced random walk and RWRE on trees, Ann. Probab., Volume 16 (1988) no. 3, pp. 1229-1241 | Article

[34] Rassoul-Agha, Firas; Seppäläinen, Timo Almost sure functional central limit theorem for ballistic random walk in random environment, Ann. Inst. Henri Poincar�, Probab. Stat., Volume 45 (2009) no. 2, pp. 373-420 | Article

[35] Sabot, Christophe Random walks in random Dirichlet environment are transient in dimension d3, Probab. Theory Relat. Fields, Volume 151 (2011) no. 1–2, pp. 297-317 (https://arxiv.org/abs/0811.4285) | Article

[36] Sabot, Christophe Random Dirichlet environment viewed from the particle in dimension d3, Ann. Probab., Volume 41 (2013) no. 2, pp. 722-743 | Article

[37] Sabot, Christophe; Tarrès, Pierre Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model, J. Eur. Math. Soc. (JEMS), Volume 17 (2015) no. 9, pp. 2353-2378 | Article

[38] Sabot, Christophe; Tournier, Laurent Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment, Ann. Inst. Henri Poincar�, Probab. Stat., Volume 47 (2011) no. 1, pp. 1-8 | Article

[39] Seppäläinen, Timo Scaling for a one-dimensional directed polymer with boundary conditions, Ann. Probab., Volume 40 (2012) no. 1, pp. 19-73 | Article

[40] Simenhaus, François Asymptotic direction for random walks in random environments, Ann. Inst. Henri Poincar�, Probab. Stat., Volume 43 (2007) no. 6, pp. 751-761 | Article

[41] Solomon, Fred Random walks in a random environment, Ann. Probab., Volume 3 (1975), pp. 1-31 | Article

[42] Sznitman, Alain-Sol Slowdown estimates and central limit theorem for random walks in random environment, J. Europ. Math. Soc., Volume 2 (2000) no. 2, pp. 93-143 | Article

[43] Sznitman, Alain-Sol On a class of transient random walks in random environment, Ann. Probab., Volume 29 (2001) no. 2, pp. 724-765 | Article

[44] Sznitman, Alain-Sol; Zeitouni, Ofer An invariance principle for isotropic diffusions in random environment, Invent. Math., Volume 164 (2006) no. 3, pp. 455-567 | Article

[45] Sznitman, Alain-Sol; Zerner, Martin P. W. A law of large numbers for random walks in random environment, Ann. Probab., Volume 27 (1999) no. 4, pp. 1851-1869 | Article

[46] Tournier, Laurent Integrability of exit times and ballisticity for random walks in Dirichlet environment, Electron. J. Probab., Volume 14 (2009), pp. 431-451 (electronic only) | Article

[47] Tournier, Laurent Asymptotic direction of random walks in Dirichlet environment, Ann. Inst. Henri Poincar�, Probab. Stat., Volume 51 (2015) no. 2, pp. 716-726 | Article

[48] Zeitouni, Ofer Random walks in random environment, Lectures on probability theory and statistics: École d’été de probabilités de Saint-Flour XXXI-2001 (Lect. Notes Math.), Volume 1837, Springer, 2004, pp. 191-312

[49] Zerner, Martin P. W. A non-ballistic law of large numbers for random walks in i.i.d. random environment, Electron. Comm. Probab., Volume 7 (2002), pp. 191-197 (paper no. 19, electronic only) | Article

[50] Zerner, Martin P. W.; Merkl, Franz A zero-one law for planar random walks in random environment, Ann. Probab., Volume 29 (2001) no. 4, pp. 1716-1732