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Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment

Jonathon Peterson — 2009

Annales de l'I.H.P. Probabilités et statistiques

We consider a nearest-neighbor, one-dimensional random walk { } in a random i.i.d. environment, in the regime where the walk is transient with speed >0 and there exists an ∈(1, 2) such that the annealed law of ( − ) converges to a stable law of parameter . Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences { ...

Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

Jonathon Peterson; Gennady Samorodnitsky — 2013

Annales de l'I.H.P. Probabilités et statistiques

We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $κ g t; 0$ that determines the fluctuations of the process. When $0 l t; κ l t; 2$ , the averaged distributions of the hitting times of the random walk converge to a $κ$ -stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for...

Maximal displacement for bridges of random walks in a random environment

Nina Gantert; Jonathon Peterson — 2011

Annales de l'I.H.P. Probabilités et statistiques

It is well known that the distribution of simple random walks on ℤ conditioned on returning to the origin after 2 steps does not depend on =(1=1), the probability of moving to the right. Moreover, conditioned on {2=0} the maximal displacement max≤2| | converges in distribution when scaled by √ (diffusive scaling). We consider the analogous problem for transient random walks in random environments on ℤ. We show that under the quenched law (conditioned on the environment...

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Currently displaying 1 – 3 of 3

Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment

Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

Maximal displacement for bridges of random walks in a random environment