Random walk in random scenery and self-intersection local times in dimensions \(d \geq 5\). (English) Zbl 1116.60057

Summary: Let \(\{S_{k}\), \(k \geq 0\}\) be a symmetric random walk on \({\mathbb Z}^d\), and \(\{\eta(x), x\in {\mathbb Z^d\}}\) an independent random field of centered i.i.d. random variables with tail decay \(P(\eta(x)> t)\approx\exp(-t^{\alpha})\). We consider a random walk in random scenery, that is \(X_n = \eta (S_0) +\cdots+\eta(S_n)\). We present asymptotics for the probability, over both randomness, that \(X_{n} > n^{\beta}\) for \(\beta > 1/2\) and \(\alpha > 1\). To obtain such asymptotics, we establish large deviation estimates for the self-intersection local times process \(\sum_x l_n^2(x)\), where \(l_{n}(x)\) is the number of visits of site \(x\) up to time \(n\).


60K37 Processes in random environments
60F10 Large deviations
60J55 Local time and additive functionals
60G50 Sums of independent random variables; random walks
Full Text: DOI arXiv


[1] Asselah, A.: Large deviations estimates for self-intersection local times for simple random walk in \(\mathbb{Z}^3\). On the arXiv: math.PR/0602074, (Preprint 2006) · Zbl 1135.60340
[2] Asselah, A.; Castell, F., Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields, 126, 4, 497-527 (2003) · Zbl 1043.60018 · doi:10.1007/s00440-003-0265-3
[3] Asselah, A., Castell, F.: A note on random walk in random scenery. To appear in Annales de l’I.H.P. (2006) · Zbl 1112.60088
[4] Bass, R. F.; Chen, X., Self-intersection local time: critical exponent, large deviations, and laws of the iterated logarithm, Ann. Probab., 32, 4, 3221-3247 (2004) · Zbl 1075.60097 · doi:10.1214/009117904000000504
[5] Bass, R.F., Chen, X., Rosen, J.: Moderate deviations and laws of the iterated logarithm for the renormalized self-intersection local times of planar random walks. arXiv: math.PR/0506414, (Preprint 2005) · Zbl 1112.60016
[6] Borodin, A. N., Limit Theorems for sums of independent random variables defined on a transient random walk, Investigations in the theory of probability distributions, IV. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 85, 17-29 (1979) · Zbl 0417.60027
[7] Borodin, A. N., A limit Theorem for sums of independent random variables defined on a recurrent random walk, Dokl. Akad. Nauk. SSSR, 246, 4, 786-787 (1979) · Zbl 0423.60025
[8] Castell, F., Moderate deviations for diffusions in a random Gaussian shear flow drift, Ann. Inst. H. Poincaré Probab. Statist., 40, 3, 337-366 (2004) · Zbl 1042.60009 · doi:10.1016/S0246-0203(03)00067-0
[9] Castell, F.; Pradeilles, F., Annealed large deviations for diffusions in a random Gaussian shear flow drift, Stochastic Process. Appl., 94, 171-197 (2001) · Zbl 1051.60028 · doi:10.1016/S0304-4149(01)00081-3
[10] Chen, X., Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab., 32, 4, 3248-3300 (2004) · Zbl 1067.60071 · doi:10.1214/009117904000000513
[11] Chen, X.; Li, W. V., Large and moderate deviations for intersection local times, Probab. Theory Related Fields, 128, 2, 213-254 (2004) · Zbl 1038.60074 · doi:10.1007/s00440-003-0298-7
[12] Donsker, M. D.; Varadhan, S. R.S., On the number of distinct sites visited by a random walk, Comm. Pure Appl. Math., 32, 6, 721-747 (1979) · Zbl 0418.60074
[13] Gantert, N., van der Hofstad, R., König, W.: Deviations of a random walk in a random scenery with stretched exponential tails. arXiv:math.PR/0411361, (Preprint 2004) · Zbl 1100.60056
[14] Gantert, N., König, W., Shi, Z.: Annealed deviations of random walk in random scenery. arXiv.:math.PR/0408327, (Preprint 2004) · Zbl 1119.60083
[15] Kasahara, Y., Tauberian theorems of exponential type, J. Math. Kyoto Univ., 18, 2, 209-219 (1978) · Zbl 0421.40009
[16] Kesten, H.; Spitzer, F., A limit theorem related to a new class of self-similar processes, Z. Wahrsch. Verw. Gebiete, 50, 1, 5-25 (1979) · Zbl 0396.60037 · doi:10.1007/BF00535672
[17] Khanin, K.M., Mazel, A.E., Shlosman S.B., Sinai, Y.A.G.: Loop condensation effects in the behavior of random walks. The Dynkin Festschrift, pp. 167-184. Prog. Probab., vol. 34, Birkhäuser Boston, Boston (1994). · Zbl 0814.60063
[18] Le Gall, J.F.: Sur le temps local d’intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan. Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., 1123, pp. 314-331. Springer, Berlin Heidelberg New York (1985). · Zbl 0563.60072
[19] Mansmann, U., The free energy of the Dirac polaron, an explicit solution, Stochastics Stochastics Rep., 34, 1-2, 93-125 (1991) · Zbl 0726.60021
[20] Matheron, G.; de Marsily, G., Is transport in porous media always diffusive? A counterexample, Water Resources Res., 16, 901-907 (1980)
[21] Westwater, M. J., On Edwards’ model for long polymer chains, Commun. Math. Phys., 72, 2, 131-174 (1980) · Zbl 0431.60100 · doi:10.1007/BF01197632
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.