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For a homogeneous random walk in the quarter plane with nearest-neighbortransitions, starting from some state $(i_0,j_0)$, we study the event that thewalk reaches the vertical axis, before reaching the horizontal axis. We derivean exact expression for the probability of this event, and derive an asymptoticexpression for the case when $i_0$ becomes large, a situation in which theevent becomes highly unlikely. The exact expression follows from the solutionof a boundary value problem and is in terms of an integral that involves aconformal gluing function. The asymptotic expression follows from theasymptotic evaluation of this integral. Our results find applications in amodel for nucleosome shifting, the voter model and the asymmetric exclusionprocess.

Random walks reaching against all odds the other side of the quarter plane