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Random Walks Reaching Against all Odds the other Side of the Quarter Plane

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Miscellaneous topics of analysis in the complex domain Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Johan S. H. van Leeuwaarden  and
Kilian Raschel
Johan S. H. van Leeuwaarden*
Affiliation:
Eindhoven University of Technology
Kilian Raschel*
Affiliation:
CNRS and Université de Tours
*
∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗ Postal address: CNRS and Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France. Email address: [email protected]
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Abstract

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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.

Keywords

Random walks in the quarter planehitting probability of the boundarynucleosome shiftingvoter modelasymmetric exclusion process

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 82C22: Interacting particle systems 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 85 - 102
Copyright
Copyright © Applied Probability Trust 2013 

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