Maximal Displacement for Bridges of Random Walks in a Random Environment

TL;DR

This paper investigates the maximal displacement of one-dimensional random walks in random environments conditioned to return to the origin, revealing different scaling behaviors depending on the environment's drift characteristics.

Contribution

It extends classical results by analyzing the maximal displacement in transient environments, providing precise asymptotics and scaling laws for various types of environments.

Findings

Maximal displacement scales as n^{κ/(κ+1)} in nestling environments.

In non-nestling environments, maximal displacement is between n^{1-ε} and n/(\,ln n)^{2-ε}.

Decay rate of return probability P_ω(X_{2n}=0) is explicitly characterized.

Abstract

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 $P_\omega$ (conditioned on the environment $\omega$), 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 $n^{\kappa/(\kappa+1)}$, where the constant $\kappa>0$ 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 $P_\omega(X_{2n}=0)$. In particular, for certain non-nestling environments we show that $P_\omega(X_{2n}=0) = \exp\{-Cn -C'n/(\ln n)^2 + o(n/(\ln n)^2) \}$ with explicit constants $C,C'>0$.