 
Summary: AN ANALYTICAL EXPRESSION FOR CROSSING PATH PROBABILITIES
IN 2D RANDOM WALKS
MARC ARTZROUNI
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF PAU; 64000 PAU; FRANCE
Abstract. We investigate crossing path probabilities for two agents that move randomly in a bounded
region of the plane or on a sphere (denoted R). At each discrete timestep the agents move, independently,
fixed distances d1 and d2 at angles that are uniformly distributed in (0, 2). If R is large enough and the
initial positions of the agents are uniformly distributed in R, then the probability of paths crossing at the
first timestep is close to 2d1d2/(A[R]), where A[R] is the area of R. Simulations suggest that the longrun
rate at which paths cross is also close to 2d1d2/(A[R]) (despite marked departures from uniformity and
independence conditions needed for such a conclusion).
Keywords: random walk, central limit theorem, intersections
AMS Classification: 82B41, 82C41, 60G50
1. Introduction. Random walks have been studied in abstract settings such as integer
lattices Zd
or Riemannian manifolds ([1], [5], [8], [11]). In applied settings there are many
spatially explicit individualbased models (IBMs) in which the behavior of the system is
determined by the meeting of randomly moving agents. The transmission of a pathogenic
agent, the spread of a rumor, or the sharing of some property when randomly moving particles
