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Large deviations for random walks on Galton–Watson trees: averaging and uncertainty
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  • Published: February 2002

Large deviations for random walks on Galton–Watson trees: averaging and uncertainty

  • Amir Dembo1,
  • Nina Gantert2,
  • Yuval Peres3 &
  • …
  • Ofer Zeitouni4 

Probability Theory and Related Fields volume 122, pages 241–288 (2002)Cite this article

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Abstract.

 In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk {X n } on a Galton–Watson tree T, i.e., on the family tree arising from a supercritical branching process. Denote by |X n | the distance between the node X n and the root of T. Our main result is the almost sure equality of the large deviation rate function for |X n |/n under the “quenched measure” (conditional upon T), and the rate function for the same ratio under the “annealed measure” (averaging on T according to the Galton–Watson distribution). This equality hinges on a concentration of measure phenomenon for the momentum of the walk. (The momentum at level n, for a specific tree T, is the average, over random walk paths, of the forward drift at the hitting point of that level). This concentration, or certainty, is a consequence of the uncertainty in the location of the hitting point. We also obtain similar results when {X n } is a λ-biased walk on a Galton–Watson tree, even though in that case there is no known formula for the asymptotic speed. Our arguments rely at several points on a “ubiquity” lemma for Galton–Watson trees, due to Grimmett and Kesten (1984).

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Authors and Affiliations

  1. Departments of Mathematics Research and of Statistics, Stanford University, Stanford, CA 94305, USA. e-mail: amir@math.stanford.edu. Research partially supported by NSF grant #DMS-0072331 and by a US-Israel BSF grant., , , , , , US

    Amir Dembo

  2. Mathematics Department, Karlsruhe University, D-76128 Karlsruhe, Germany. e-mail: N.Gantert@math.uni-karlsruhe.de. Research partially supported by the DFG., , , , , , DE

    Nina Gantert

  3. Department of Statistics, UC Berkeley, Berkeley, CA 94720, USA and Institute of Mathematics, Hebrew University, Jerusalem, Israel. e-mail: peres@stat.berkeley.edu. Research partially supported by NSF grant #DMS-9803597 and by a US-Israel BSF grant., , , , , , IL

    Yuval Peres

  4. Department of Electrical Engineering, Technion, Haifa 32000, Israel. e-mail: zeitouni@ee.technion.ac.il. Research partially supported by a US- Israel BSF grant and by the fund for promotion of research at the Technion., , , , , , IL

    Ofer Zeitouni

Authors
  1. Amir Dembo
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  2. Nina Gantert
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  3. Yuval Peres
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  4. Ofer Zeitouni
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Additional information

Received: 15 November 2000 / Revised version: 27 February 2001 / Published online: 19 December 2001

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Dembo, A., Gantert, N., Peres, Y. et al. Large deviations for random walks on Galton–Watson trees: averaging and uncertainty. Probab Theory Relat Fields 122, 241–288 (2002). https://doi.org/10.1007/s004400100162

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  • Issue Date: February 2002

  • DOI: https://doi.org/10.1007/s004400100162

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Keywords

  • Rate Function
  • Random Walk
  • Specific Tree
  • Deviation Rate
  • Random Environment
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