Random walks reaching against all odds the other side of the quarter plane
Abstract
For a homogeneous random walk in the quarter plane with nearestneighbor transitions, starting from some state $(i_0,j_0)$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact expression for the probability of this event, and derive an asymptotic expression for the case when $i_0$ becomes large, a situation in which the event becomes highly unlikely. The exact expression follows from the solution of a boundary value problem and is in terms of an integral that involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model and the asymmetric exclusion process.
 Publication:

arXiv eprints
 Pub Date:
 April 2011
 arXiv:
 arXiv:1104.3034
 Bibcode:
 2011arXiv1104.3034V
 Keywords:

 Mathematics  Probability
 EPrint:
 18 pages, 2 figures, to appear in Journal of Applied Probability