Random walks in Dirichlet environment: an overview

Christophe Sabot; Laurent Tournier

doi:10.5802/afst.1542

Christophe Sabot ¹ ; Laurent Tournier ²

¹ Universitï¿½ Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43, bd du 11 novembre, 69622 Villeurbanne cedex, France
² Universitï¿½ Paris 13, Sorbonne Paris Citï¿½, LAGA, CNRS UMR 7539, 93430 Villetaneuse, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 463-509.

Résumé
Abstract

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 ${(α_{i})}_{i = 1, ..., 2 d}$ , un pour chaque direction dirigée de $ℤ^{d}$ . 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 $ℤ^{d}$ 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 ${(α_{i})}_{i = 1, ..., 2 d}$ , one for each oriented direction of $ℤ^{d}$ . 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 : 2017-04-12

DOI : 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

Affiliations des auteurs :

Christophe Sabot ¹ ; Laurent Tournier ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AFST_2017_6_26_2_463_0,
     author = {Christophe Sabot and Laurent Tournier},
     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 = {https://afst.centre-mersenne.org/articles/10.5802/afst.1542/}
}

TY  - JOUR
AU  - Christophe Sabot
AU  - Laurent Tournier
TI  - Random walks in Dirichlet environment: an overview
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2017
SP  - 463
EP  - 509
VL  - 26
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1542/
DO  - 10.5802/afst.1542
LA  - en
ID  - AFST_2017_6_26_2_463_0
ER  -

%0 Journal Article
%A Christophe Sabot
%A Laurent Tournier
%T Random walks in Dirichlet environment: an overview
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2017
%P 463-509
%V 26
%N 2
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1542/
%R 10.5802/afst.1542
%G en
%F AFST_2017_6_26_2_463_0

Christophe Sabot; Laurent Tournier. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1542/

Bibliographie
Cité par

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

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

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

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

[6] Noam Berger; Ofer Zeitouni 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] Erwin Bolthausen; Alain-Sol Sznitman Ten lectures on random media, DMV Seminar, 32, Birkhäuser, 2002, vi+116 pages | DOI

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

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

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

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

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

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

[14] Francis Comets; Nina Gantert; Ofer Zeitouni 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) | DOI

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

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

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

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

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

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

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

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

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

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

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

[26] Michael S. Keane; Silke W. W. Rolles 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] Harry Kesten Random difference equations and renewal theory for products of random matrices, Acta Mathematica, Volume 131 (1973) no. 1, pp. 207-248 | DOI

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

[29] Gregory F. Lawler 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 | DOI

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

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

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

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

[34] Firas Rassoul-Agha; Timo Seppäläinen Almost sure functional central limit theorem for ballistic random walk in random environment, Ann. Inst. Henri Poincarï¿½, Probab. Stat., Volume 45 (2009) no. 2, pp. 373-420 | DOI

[35] Christophe Sabot Random walks in random Dirichlet environment are transient in dimension $d \geq 3$ , Probab. Theory Relat. Fields, Volume 151 (2011) no. 1–2, pp. 297-317 (https://arxiv.org/abs/0811.4285) | DOI

[36] Christophe Sabot Random Dirichlet environment viewed from the particle in dimension $d \geq 3$ , Ann. Probab., Volume 41 (2013) no. 2, pp. 722-743 | DOI

[37] Christophe Sabot; Pierre Tarrès 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 | DOI

[38] Christophe Sabot; Laurent Tournier Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment, Ann. Inst. Henri Poincarï¿½, Probab. Stat., Volume 47 (2011) no. 1, pp. 1-8 | DOI

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

[40] François Simenhaus Asymptotic direction for random walks in random environments, Ann. Inst. Henri Poincarï¿½, Probab. Stat., Volume 43 (2007) no. 6, pp. 751-761 | DOI

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

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

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

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

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

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

[47] Laurent Tournier Asymptotic direction of random walks in Dirichlet environment, Ann. Inst. Henri Poincarï¿½, Probab. Stat., Volume 51 (2015) no. 2, pp. 716-726 | DOI

[48] Ofer Zeitouni 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] Martin P. W. Zerner 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) | DOI

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

Cité par Sources :