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 time-step the agents move, independently, fixed distances d 1 and d 2 at angles that are uniformly distributed in (0, 2π). 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 time-step is close to 2d 1 d 2 /(πA[R]), where A[R] is the area of R. Simulations suggest that the long-run rate at which paths cross is also close to 2d 1 d 2 /(πA[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 Z d or Riemannian manifolds ([1], [5], [8], [11]). In applied settings there are many spatially explicit individual-based 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 meet are examples that come to mind in biology, sociology, or physics ([4], [9], [7], [6], [3]). Although IBMs are powerful tools for the description of complex systems, they suffer from a shortage of analytical results. For example, if a susceptible and an infective agent move randomly in some bounded space, what is the probability of them meeting, and hence of the transmission of the infection? What is the average time until the meeting takes place? Also, in many of these models the movement of agents is conceptualized as discrete transitions between square or hexagonal cells ([4]). However, such a stylized representation of individual movements may not always be entirely realistic. In the present paper we begin to address these issues by considering a random walk in a bounded region of the plane or on the sphere, which we denote by R. The model evolves in discrete time. At each time-step an agent leaves its current position at a uniformly distributed angle in the (0, 2π) interval. On the plane the agent moves a fixed distance d in a straight line. On a sphere the agent moves a fixed distance d on a geodesic. Here we will consider two such agents who move different distances d 1 and d 2 at each time-step. We assume that the agents’ initial positions are uniformly distributed in R. The 1