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Maximal Displacement for Bridges of Random Walks in a Random Environment

It is well known that the distribution of simple random walks on $\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on $\{S_{2n}=0\}$ the maximal displacement $\max_{k\leq 2n} |S_k|$ converges in distribution when scaled by $\sqrt{n}$ (diffusive scaling). We consider the analogous problem for transient random walks in random environments on $\bf{Z}$. 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 $\sqrt{n}$. 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 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 $n^{1-\varepsilon}$ and at most $n/(\ln n)^{2-\varepsilon}$ for any $\varepsilon>0$. 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 $C,C'>0$.