A quenched functional central limit theorem for planar random walks in random sceneries

Nadine Guillotin-Plantard (Université Lyon 1)
Julien Poisat (Leiden Universiteit)
Renato Soares dos Santos (Université Lyon 1)


Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $\xi = (\xi(x))_{x \in \mathbb{Z}^d}$ of i.i.d. random variables, which is called the random scenery, and a random walk $S = (S_n)_{n \in \mathbb{N}}$ evolving in $\mathbb{Z}^d$, independent of the scenery. The RWRS $Z = (Z_n)_{n \in \mathbb{N}}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n \xi(S_k)$. The law of $Z$ under the joint law of $\xi$ and $S$ is called "annealed'', and the conditional law given $\xi$ is called "quenched''. Recently, functional central limit theorems under the quenched law were proved for $Z$ by the first two authors for a class of transient random walks including walks with finite variance in dimension $d \ge 3$. In this paper we extend their results to dimension $d=2$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-9

Publication Date: January 28, 2014

DOI: 10.1214/ECP.v19-3002


  • Asselah, Amine; Castell, Fabienne. Random walk in random scenery and self-intersection local times in dimensions $d\ge5$. Probab. Theory Related Fields 138 (2007), no. 1-2, 1--32. MR2288063
  • Ben Arous, Gérard; Černý, Jiří. Scaling limit for trap models on $\Bbb Z^ d$. Ann. Probab. 35 (2007), no. 6, 2356--2384. MR2353391
  • Berger, Noam; Zeitouni, Ofer. A quenched invariance principle for certain ballistic random walks in i.i.d. environments. In and out of equilibrium. 2, 137--160, Progr. Probab., 60, Birkhäuser, Basel, 2008. MR2477380
  • Bolthausen, Erwin. A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989), no. 1, 108--115. MR0972774
  • Bolthausen, Erwin; Sznitman, Alain-Sol. On the static and dynamic points of view for certain random walks in random environment. Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. Methods Appl. Anal. 9 (2002), no. 3, 345--375. MR2023130
  • Borodin, A. N. A limit theorem for sums of independent random variables defined on a recurrent random walk. (Russian) Dokl. Akad. Nauk SSSR 246 (1979), no. 4, 786--787. MR0543530
  • Borodin, A. N. Limit theorems for sums of independent random variables defined on a transient random walk. (Russian) Investigations in the theory of probability distributions, IV. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 (1979), 17--29, 237, 244. MR0535455
  • Campanino, M.; Petritis, D. Random walks on randomly oriented lattices. Markov Process. Related Fields 9 (2003), no. 3, 391--412. MR2028220
  • Castell, Fabienne. Moderate deviations for diffusions in a random Gaussian shear flow drift. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 3, 337--366. MR2060457
  • F. Castell, N. Guillotin-Plantard, and F. Pène, Limit theorems for one and two-dimensional random walks in random scenery, Ann. Inst. Henri Poincaré Probab. Statist. 49 (2) (2013) 506--528.
  • Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise; Schapira, Bruno. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Ann. Probab. 39 (2011), no. 6, 2079--2118. MR2932665
  • Castell, F.; Pradeilles, F. Annealed large deviations for diffusions in a random Gaussian shear flow drift. Stochastic Process. Appl. 94 (2001), no. 2, 171--197. MR1840830
  • Cohen, Serge; Dombry, Clément. Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions. J. Math. Kyoto Univ. 49 (2009), no. 2, 267--286. MR2571841
  • Csáki, Endre; König, Wolfgang; Shi, Zhan. An embedding for the Kesten-Spitzer random walk in random scenery. Stochastic Process. Appl. 82 (1999), no. 2, 283--292. MR1700010
  • Csáki, E.; Révész, P. Strong invariance for local times. Z. Wahrsch. Verw. Gebiete 62 (1983), no. 2, 263--278. MR0688990
  • Le Doussal, Pierre. Diffusion in layered random flows, polymers, electrons in random potentials, and spin depolarization in random fields. J. Statist. Phys. 69 (1992), no. 5-6, 917--954. MR1192029
  • den Hollander, Frank; Steif, Jeffrey E. Random walk in random scenery: a survey of some recent results. Dynamics & stochastics, 53--65, IMS Lecture Notes Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, 2006. MR2306188
  • Fleischmann, Klaus; Mörters, Peter; Wachtel, Vitali. Moderate deviations for a random walk in random scenery. Stochastic Process. Appl. 118 (2008), no. 10, 1768--1802. MR2454464
  • Gantert, Nina; König, Wolfgang; Shi, Zhan. Annealed deviations of random walk in random scenery. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 1, 47--76. MR2288269
  • Guillotin-Plantard, Nadine; Poisat, Julien. Quenched central limit theorems for random walks in random scenery. Stochastic Process. Appl. 123 (2013), no. 4, 1348--1367. MR3016226
  • Guillotin-Plantard, Nadine; Prieur, Clémentine. Limit theorem for random walk in weakly dependent random scenery. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 4, 1178--1194. MR2744890
  • Kesten, H.; Spitzer, F. A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 5--25. MR0550121
  • Khoshnevisan, Davar; Lewis, Thomas M. A law of the iterated logarithm for stable processes in random scenery. Stochastic Process. Appl. 74 (1998), no. 1, 89--121. MR1624017
  • Le Gall, Jean-François; Rosen, Jay. The range of stable random walks. Ann. Probab. 19 (1991), no. 2, 650--705. MR1106281
  • G. Matheron and G. de Marsily, Is Transport in Porous Media Always Diffusive? A Counterexample, Water Resources Research, vol. 16, no. 5 (1980) 901-917.
  • de la Peña, Victor H. A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27 (1999), no. 1, 537--564. MR1681153
  • Rassoul-Agha, Firas; Seppäläinen, Timo. Almost sure functional central limit theorem for ballistic random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 2, 373--420. MR2521407
  • Spitzer, Frank. Principles of random walk. Second edition. Graduate Texts in Mathematics, Vol. 34. Springer-Verlag, New York-Heidelberg, 1976. xiii+408 pp. MR0388547

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.