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Random-walk in Beta-distributed random environment

Abstract

We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in \(\mathbb {Z}\) which performs nearest neighbour jumps with transition probabilities drawn according to the Beta distribution. We also describe a related directed polymer model, which is a limit of the q-Hahn interacting particle system. Using a Fredholm determinant representation for the quenched probability distribution function of the walker’s position, we are able to prove second order cube-root scale corrections to the large deviation principle satisfied by the walker’s position, with convergence to the Tracy–Widom distribution. We also show that this limit theorem can be interpreted in terms of the maximum of strongly correlated random variables: the positions of independent walkers in the same environment. The zero-temperature counterpart of the Beta RWRE can be studied in a parallel way. We also prove a Tracy–Widom limit theorem for this model.

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Notes

  1. 1.

    By Crámer’s Theorem, it is the Legendre transform of \(z\mapsto \log \left( \frac{e^{-z}+e^{z}}{2}\right) \). One finds

    $$\begin{aligned} I^a(x)= {\left\{ \begin{array}{ll} \frac{1}{2} \big ( (1+x)\log (1+x)+ (1-x)\log (1-x)\big )\text { for }x\in [-1,1],\\ +\infty \text { else.} \end{array}\right. } \end{aligned}$$

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Acknowledgments

G.B. would like to thank Vu-Lan Nguyen for interesting discussions. G.B. and I.C. thank Firas Rassoul-Agha and Timo Seppäläinen for useful comments on a first version of the paper. G.B. was partially supported by the Laboratoire de Probabilités et Modèles Aléatoires UMR CNRS 7599, Université Paris-Diderot–Paris 7, as well as the Packard Foundation through I.C.’s Packard Fellowship for Science and Engineering. I.C. was partially supported by the NSF through DMS-1208998, the Clay Mathematics Institute through a Clay Research Fellowship, the Institute Henri Poincaré through the Poincaré Chair, and the Packard Foundation through a Packard Fellowship for Science and Engineering.

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Correspondence to Guillaume Barraquand.

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Barraquand, G., Corwin, I. Random-walk in Beta-distributed random environment. Probab. Theory Relat. Fields 167, 1057–1116 (2017). https://doi.org/10.1007/s00440-016-0699-z

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Mathematics Subject Classification

  • 60K37
  • 60F10
  • 60G70
  • 60K35
  • 82B23