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

  • Johan S. H. van Leeuwaarden (a1) and Kilian Raschel (a2)

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.

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Copyright

Corresponding author

Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: j.s.h.v.leeuwaarden@tue.nl
∗∗ Postal address: CNRS and Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France. Email address: kilian.raschel@lmpt.univ-tours.fr

References

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[1] Aspandiiarov, S., Iasnogorodski, R. and Menshikov, M. (1996). Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant. Ann. Prob. 24, 932960.
[2] Belitsky, V., Ferrari, P. A., Menshikov, M. V. and Popov, S. Y. (2001). A mixture of the exclusion process and the voter model. Bernoulli 7, 119144.
[3] Durrett, R. and Levin, S. A. (1994). Stochastic spatial models: a user's guide to ecological applications. Phil. Trans. R. Soc. London B 343, 329350.
[4] Fayolle, G. and Raschel, K. (2011). Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group. Markov Process. Relat. Fields 17, 619636.
[5] Fayolle, G., Iasnogorodski, R. and Malyshev, V. (1999). Random Walks in the Quarter-Plane. Springer, Berlin.
[6] Godrèche, C. et al. (1995). Spontaneous symmetry breaking: exact results for a biased random walk model of an exclusion process. J. Phys. A 28, 60396071.
[7] Jones, G. A. and Singerman, D. (1987). Complex Functions. Cambridge University Press.
[8] Kurkova, I. and Raschel, K. (2011). Random walks in Z2 with non-zero drift absorbed at the axes. Bull. Soc. Math. France 139, 341387.
[9] Litvinchuk, G. S. (2000). Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Kluwer Academic, Dordrecht.
[10] Malyshev, V. (1971). Positive random walks and Galois theory. Uspekhi Mat. Nauk 26, 227228.
[11] Raschel, K. (2012). Counting walks in a quadrant: a unified approach via boundary value problems. J. Europ. Math. Soc. 14, 749777.
[12] Van Opheusden, S. C. F. and Redig, F. (2010). Markov models for nucleosome dynamics during transcription: breathing and sliding. Bachelor Thesis, Leiden University.

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

  • Johan S. H. van Leeuwaarden (a1) and Kilian Raschel (a2)

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