## The Annals of Probability

### Random Dirichlet environment viewed from the particle in dimension $d\ge3$

Christophe Sabot

#### Abstract

We consider random walks in random Dirichlet environment (RWDE), which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${\mathbb{Z}}^{d}$, RWDE are parameterized by a $2d$-tuple of positive reals called weights. In this paper, we characterize for $d\ge3$ the weights for which there exists an absolutely continuous invariant probability distribution for the process viewed from the particle. We can deduce from this result and from [Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8] a complete description of the ballistic regime for $d\ge3$.

#### Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 722-743.

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.aop/1362750940

Digital Object Identifier
doi:10.1214/11-AOP699

Mathematical Reviews number (MathSciNet)
MR3077524

Zentralblatt MATH identifier
1269.60077

#### Citation

Sabot, Christophe. Random Dirichlet environment viewed from the particle in dimension $d\ge3$. Ann. Probab. 41 (2013), no. 2, 722--743. doi:10.1214/11-AOP699. https://projecteuclid.org/euclid.aop/1362750940

#### References

• [1] Berger, N. (2008). Limiting velocity of high-dimensional random walk in random environment. Ann. Probab. 36 728–738.
• [2] Berger, N. and Zeitouni, O. (2008). A quenched invariance principle for certain ballistic random walks in i.i.d. environments. In In and Out of Equilibrium. 2. Progress in Probability 60 137–160. Birkhäuser, Basel.
• [3] Bolthausen, E. and Sznitman, A.-S. (2002). Ten Lectures on Random Media. DMV Seminar 32. Birkhäuser, Basel.
• [4] Bolthausen, E. and Sznitman, A.-S. (2002). On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 345–375.
• [5] Bolthausen, E. and Zeitouni, O. (2007). Multiscale analysis of exit distributions for random walks in random environments. Probab. Theory Related Fields 138 581–645.
• [6] Bricmont, J. and Kupiainen, A. (1991). Random walks in asymmetric random environments. Comm. Math. Phys. 142 345–420.
• [7] Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
• [8] Enriquez, N. and Sabot, C. (2006). Random walks in a Dirichlet environment. Electron. J. Probab. 11 802–817 (electronic).
• [9] Enriquez, N., Sabot, C. and Zindy, O. (2009). Limit laws for transient random walks in random environment on $\mathbb{Z}$. Ann. Inst. Fourier (Grenoble) 59 2469–2508.
• [10] Ford, L. R. Jr. and Fulkerson, D. R. (1962). Flows in Networks. Princeton Univ. Press, Princeton, NJ.
• [11] Kalikow, S. A. (1981). Generalized random walk in a random environment. Ann. Probab. 9 753–768.
• [12] Kesten, H. (1977). A renewal theorem for random walk in a random environment. In Probability (Proc. Sympos. Pure Math., Vol. XXXI, Univ. Illinois, Urbana, Ill., 1976) 67–77. Amer. Math. Soc., Providence, RI.
• [13] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
• [14] Komorowski, T., Olla, S. and Landim, C. Fluctuations in Markov processes. Available at http://w3.impa.br/~landim/notas.html.
• [15] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Russ. Math. Surv. 40 73–145.
• [16] Lawler, G. F. (1982/83). Weak convergence of a random walk in a random environment. Comm. Math. Phys. 87 81–87.
• [17] Lyons, R. and Peres, Y. Probabilities on trees and network. Preprint. Available at http://php.indiana.edu/~rdlyons/prbtree/prbtree.html.
• [18] Mathieu, P. (2008). Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130 1025–1046.
• [19] Molchanov, S. (1994). Lectures on random media. In Lectures on Probability Theory (Saint-Flour, 1992) (P. Bernard, ed.). Lecture Notes in Math. 1581 242–411. Springer, Berlin.
• [20] Pemantle, R. (1988). Random processes with reinforcement. Ph.D. thesis, Dept. Mathematics, MIT, Cambridge, MA.
• [21] Pemantle, R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 1229–1241.
• [22] Pemantle, R. (2007). A survey of random processes with reinforcement. Probab. Surv. 4 1–79.
• [23] Rassoul-Agha, F. (2003). The point of view of the particle on the law of large numbers for random walks in a mixing random environment. Ann. Probab. 31 1441–1463.
• [24] Rassoul-Agha, F. and Seppäläinen, T. (2009). Almost sure functional central limit theorem for ballistic random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 45 373–420.
• [25] Sabot, C. (2011). Random walks in random Dirichlet environment are transient in dimension $d\ge3$. Probab. Theory Related Fields 151 297–317.
• [26] Sabot, C. (2006). Markov chains in a Dirichlet environment and hypergeometric integrals. C. R. Math. Acad. Sci. Paris 342 57–62.
• [27] Sabot, C. and Tournier, L. (2011). Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment. Ann. Inst. Henri Poincaré Probab. Stat. 47 1–8.
• [28] Sidoravicius, V. and Sznitman, A.-S. (2004). Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 219–244.
• [29] Sznitman, A.-S. (2001). On a class of transient random walks in random environment. Ann. Probab. 29 724–765.
• [30] Sznitman, A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 509–544.
• [31] Sznitman, A.-S. and Zeitouni, O. (2006). An invariance principle for isotropic diffusions in random environment. Invent. Math. 164 455–567.
• [32] Sznitman, A.-S. and Zerner, M. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851–1869.
• [33] Tournier, L. (2009). Integrability of exit times and ballisticity for random walks in Dirichlet environment. Electron. J. Probab. 14 431–451.
• [34] Zeitouni, O. (2004). Random walks in random environment. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837 189–312. Springer, Berlin.
• [35] Zerner, M. P. W. (2002). A nonballistic law of large numbers for random walks in i.i.d. random environment. Electron. Commun. Probab. 7 191–197 (electronic).
• [36] Zerner, M. P. W. and Merkl, F. (2001). A zero–one law for planar random walks in random environment. Ann. Probab. 29 1716–1732.