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

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.

[1]  J. C. Burkill,et al.  Complex Functions , 1968, Nature.

[2]  G. Fayolle,et al.  Topics in the Constructive Theory of Countable Markov Chains: Ideology of induced chains , 1995 .

[3]  Kilian Raschel,et al.  Random walks in (Z + ) 2 with non-zero drift absorbed at the axes , 2013 .

[4]  G. S. Litvinchuk,et al.  Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift , 2000 .

[5]  G. Fayolle,et al.  Topics in the Constructive Theory of Countable Markov Chains , 1995 .

[6]  D. Mukamel,et al.  Spontaneous symmetry breaking: exact results for a biased random walk model of an exclusion process , 1995 .

[7]  Kilian Raschel,et al.  Random walks in $(\mathbb{Z}_+)^2$ with non-zero drift absorbed at the axes , 2009 .

[8]  M. Manhart,et al.  Markov Processes , 2018, Introduction to Stochastic Processes and Simulation.

[9]  Simon A. Levin,et al.  Stochastic Spatial Models: A User's Guide to Ecological Applications , 1994 .

[10]  Guy Fayolle,et al.  Random walks in the quarter plane with zero drift: an explicit criterion for the finiteness of the associated group , 2011 .

[11]  A mixture of the exclusion process and the voter model , 2000, math/0002051.

[12]  I. MacPhee,et al.  Passage-time moments and hybrid zones for the exclusion-voter model. , 2008, 0810.0392.

[13]  P. Eichelsbacher,et al.  Ordered Random Walks , 2006, math/0610850.

[14]  M. Menshikov,et al.  Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant , 1996 .

[15]  Kilian Raschel.  Counting walks in a quadrant: a unified approach via boundary value problems , 2010 .

[16]  T. Liggett.  Interacting Particle Systems , 1985 .

[17]  S. C. F. V. Opheusden.  Markov models for Nucleosome dynamics during Transcription : Breathing and Sliding , 2010 .