March 2013 Random walks reaching against all odds the other side of the quarter plane
Johan S. H. van Leeuwaarden, Kilian Raschel
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J. Appl. Probab. 50(1): 85-102 (March 2013). DOI: 10.1239/jap/1363784426


For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state (i0,j0), 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 i0 becomes large, a situation in which the event becomes highly unlikely. The integral representation follows from the solution of a boundary value problem and 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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Johan S. H. van Leeuwaarden. Kilian Raschel. "Random walks reaching against all odds the other side of the quarter plane." J. Appl. Probab. 50 (1) 85 - 102, March 2013.


Published: March 2013
First available in Project Euclid: 20 March 2013

zbMATH: 1271.60059
MathSciNet: MR3076774
Digital Object Identifier: 10.1239/jap/1363784426

Primary: 60G50
Secondary: 30E20 , 82C22

Keywords: Asymmetric exclusion process , hitting probability of the boundary , nucleosome shifting , Random walks in the quarter plane , voter model

Rights: Copyright © 2013 Applied Probability Trust


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Vol.50 • No. 1 • March 2013
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