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

Published online by Cambridge University Press:  30 January 2018

Johan S. H. van Leeuwaarden
Affiliation:
Eindhoven University of Technology
Kilian Raschel
Affiliation:
CNRS and Université de Tours
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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 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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2013 

References

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