For a homogeneous random walk in the quarter plane withnearest-neighbor transitions, starting from some state (i0,j0), westudy the event that the walk reaches the vertical axis, beforereaching the horizontal axis. We derive a certain integralrepresentation for the probability of this event, and an asymptoticexpression for the case when i0 becomes large, a situation in whichthe event becomes highly unlikely. The integral representation followsfrom the solution of a boundary value problem and involves a conformalgluing function. The asymptotic expression follows from the asymptoticevaluation of this integral. Our results find applications in a modelfor nucleosome shifting, the voter model, and the asymmetric exclusionprocess.
"Random walks reaching against all odds the other side of the quarter plane." J. Appl. Probab. 50 (1) 85 - 102, March 2013. https://doi.org/10.1239/jap/1363784426