Maximal Displacement for Bridges of Random Walks in a Random Environment

Details Maximal Displacement for Bridges of Random Walks in a Random Environment Maximal Displacement for Bridges of Random Walks in a Random Environment

2009-10-26

arxiv.org/abs/0910.4927v2

Mathematics

Probability

Revised version, 19 pages, 1 figure To appear in: AIHP Prob. & Stat

Scientific paper

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 $C,C'>0$.

Affiliated with

Also associated with

No associations

Rating

Maximal Displacement for Bridges of Random Walks in a Random Environment does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Maximal Displacement for Bridges of Random Walks in a Random Environment, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Maximal Displacement for Bridges of Random Walks in a Random Environment will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-563424