Maximal displacement for bridges of random walks in a random environment

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 {S_2n = 0} the maximal displacement max _k≤2n |S_k| 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 P_ω (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^κ/(κ+1), where the constantκ >0 depends on the law on environments. 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 n^1-ε and at most n/(ln n)^2-ε for any ε > 0. As a consequence of our proofs, we obtain precise rates of decay for P _ω(X_2n = 0). In particular, for certain non-nestling environments we show that P_ω(X_2n = 0) = exp{-Cn-C″n/(ln n)² + o(n/(ln n)²)} with explicit constants C,C″ > 0.