Maximal Displacement for Bridges of Random Walks in a Random Environment
نویسندگان
چکیده
It is well known that the distribution of simple random walks on Z conditioned on returning to the origin after 2n steps does not depend on p = P (S1 = 1), the probability of moving to the right. Moreover, conditioned on {S2n = 0} the maximal displacement maxk≤2n |Sk| converges in distribution when scaled by √ n (diffusive scaling). We consider the analogous problem for transient random walks in random environments on Z. We show that under the quenched law Pω (conditioned on the environment ω), the maximal displacement of the random walk when conditioned to return to the origin at time 2n is no longer necessarily of the order √ n. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time 2n is of order n, where the constant κ > 0 depends on the law on environment. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time 2n is at least n1−ε and at most n/(lnn)2−ε for any ε > 0. As a consequence of our proofs, we obtain precise rates of decay for Pω(X2n = 0). In particular, for certain non-nestling environments we show that Pω(X2n = 0) = exp{−Cn − C ′n/(lnn)2 + o(n/(lnn))} with explicit constants C,C ′ > 0.
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