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

Abstract : For a homogeneous random walk in the quarter plane with nearest-neighbor 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 a certain integral representation for the probability of this event, and an asymptotic expression for the case when $i_0$ becomes large, a situation in which the event becomes highly unlikely. The integral representation follows from the solution of a boundary value problem and is 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.
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Submitted on : Sunday, October 7, 2012 - 12:54:12 PM
Last modification on : Thursday, March 5, 2020 - 5:33:42 PM
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Johan van Leeuwaarden, Kilian Raschel. Random walks reaching against all odds the other side of the quarter plane. Journal of Applied Probability, Applied Probability Trust, 2013, 50 (1), pp.85-102. ⟨10.1239/jap/1363784426⟩. ⟨hal-00586295v2⟩

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