Summary
Suppose that i.i.d. random variables are attached to the edges of an infinite tree. When the tree is large enough, the partial sumsS σ along some of its infinite paths will exhibit behavior atypical for an ordinary random walk. This principle has appeared in works on branching random walks, first-passage percolation, and RWRE on trees. We establish further quantitative versions of this principle, which are applicable in these settings. In particular, different notions of speed for such a tree-indexed walk correspond to different dimension notions for trees. Finally, if the labeling variables take values in a group, then properties of the group (e.g., polynomial growth or a nontrivial Poisson boundary) are reflected in the sample-path behavior of the resulting tree-indexed walk.
Article PDF
Similar content being viewed by others
References
Aldous, D.: Probability approximations via the poisson clumping heuristic. Berlin Heidelberg New York: Springer 1989
Arnold, V.I.: mathematical methods of classical mechanics, 2nd edn. Berlin Heidelberg New York: Springer 1989
Avez, A.: Croissance des groupes de type fini et fonctions harmoniques. (Lect. Notes Math., vol. 532, pp. 35–49) Berlin Heidelberg New York: Springer 1976
Benjamini, I., Peres, Y.: Random walks on a tree and capacity in the interval. Ann. Inst. Henri Poincaré, Probab.28, 557–592 (1992)
Benjamini, I., Peres, Y.: Markov chains indexed by trees. Ann. Probab.22 (to appear 1994)
Biggins, J.D.: Chernoff's theorem in the branching random walk. J. Appl. Probab.14, 630–636 (1977)
Carleson, L.: Selected problems on exceptional sets. Princeton, New Jersey: Van Nostrand 1967
Deuschel, J.D., Stroock, D.W.: Large deviations. San Diego: Academic Press 1989
Dubins, L.E., Freedman, D.A.: Random distribution functions. (Proc. of the fifth Berkeley Symposium on Statistics and Probability) vol. II, Part I, pp. 183–214 Berkeley, Calif.: University of California Press 1967
Durrett, R.: Probability: Theory and examples. Pacific Grove, California: Wadsworth and Brooks/Cole 1991
Evans, S.: Polar and nonpolar sets for a tree-indexed process. Ann. Probab.20, 579–590 (1992)
Furstenberg, H.: Intersections of Cantor sets and transversality of semigroups. In: Gunning, R.C. (ed.) Problems in Analysis, a Symposium in honor of S. Bochner. Princeton, New Jersey: Princeton University Press 1970
Hebisch, W., Saloff-Coste, L.: Gaussian estimates for Markov chains and random walks on groups. Ann. Probab.21, 673–709 (1993)
Joffe, A., Moncayo, A.R.: Random variables, trees and branching random walks. Adv. Math.10, 401–410 (1973)
Kaimanovich, V.A., Vershik, A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab.11, 457–490 (1983)
Kesten, H.: Symmetric random walks on groups Trans. Am. Math. Soc.92, 336–354 (1959)
Kingman, J.F.C.: The first birth problem for an age-dependent branching process. Ann. Probab.12, 341–345 (1975)
Ledrappier, F.: Sharp estimates for the entropy. (Preprint 1992)
Lyons, R.: The Ising model and percolation on trees and tree-like graphs. Commun. Math. Phys.125, 337–353 (1989)
Lyons, R.: Random walks and percolation on trees. Ann. Probab.18, 931–958 (1990)
Lyons, R.: Random walks, capacity and percolation on trees. Ann. Probab.20, 2043–2088 (1992)
Lyons, R., Pemantle, R.: Random walk in a random environment and first passage percolation on trees. Ann. Probab.20, 125–136 (1992)
Pemantle R., Peres, Y.: Critical random walk in random environment on trees (Preprint 1993)
Ritter, G.: Growth of random walk conditioned to stay positive. Ann. Probab.9, 699–704 (1981)
Spitzer, F.: Principles of Random Walk. Princeton, New Jersey: Van Nostrand 1964
Taylor, S.J., Tricot, C.: Packing measure, and its evaluation for a Brownian path. Trans. Am. Math. Soc.288, 679–699 (1985)
Varopoulos, N.Th.: Convolution powers on locally compact groups, Bull. Sci. Math., II Sèr.111, 333–342 (1987)
Varopoulos, N.Th.: Analysis and geometry on groups; Proceedings of the I.C.M., Vol.2, pp. 951–957, Kyoto 1990
Author information
Authors and Affiliations
Additional information
Partially supported by a grant from the Landau Center for Mathematical Analysis
Partially supported by NSF grant DMS-921 3595
Rights and permissions
About this article
Cite this article
Benjamini, I., Peres, Y. Tree-indexed random walks on groups and first passage percolation. Probab. Th. Rel. Fields 98, 91–112 (1994). https://doi.org/10.1007/BF01311350
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01311350