A quenched limit theorem for the local time of random walks on \Z^2

Jürgen Gärtner; Rongfeng Sun

A quenched limit theorem for the local time of random walks on \Z^2

Probability 2008-06-10 v2

Authors: Jürgen Gärtner , Rongfeng Sun

Abstract

Let $X$ and $Y$ be two independent random walks on $\Z^2$ with zero mean and finite variances, and let $L_t(X,Y)$ be the local time of $X-Y$ at the origin at time $t$ . We show that almost surely with respect to $Y$ , $L_t(X,Y)/\log t$ conditioned on $Y$ converges in distribution to an exponential random variable with the same mean as the distributional limit of $L_t(X,Y)/\log t$ without conditioning. This question arises naturally from the study of the parabolic Anderson model with a single moving catalyst, which is closely related to a pinning model.

Keywords

quenched limit theorem local time random walks Z^2 parabolic Anderson model pinning model exponential distribution

Cite

@article{arxiv.0711.4488,
  title  = {A quenched limit theorem for the local time of random walks on \Z^2},
  author = {Jürgen Gärtner and Rongfeng Sun},
  journal= {arXiv preprint arXiv:0711.4488},
  year   = {2008}
}

Comments

To appear in Stochastic Processes and Their Applications. Updated version. 16 pages. Added discussion on d=1 and d\geq 3 as well as an open problem

A quenched limit theorem for the local time of random walks on \Z^2

Abstract

Keywords

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