Crossing paths in 2D Random Walks

1. Introduction. Random walks have been studied in abstract settings such as integer lattices Z d or Riemannian manifolds ( [4], [7], [10]). 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 ( [3], [8], [6], [5], [2]). In many of these models the movement of agents is conceptualized as discrete transitions between square or hexagonal cells ( [3]). However, such a stylized representation of individual movements may not always be entirely realistic.

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?

In the present paper we begin to answer these questions 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 paper’s central results concern the probability that the paths of the two agents cross during the first time-step. If R is either a sufficiently large bounded region of the plane or a sphere, this “first-step” probability of intersection is close to 2d 1 d 2 /(πA[R]) where A[R] is the area of R.

In applied settings we are often interested in the long-run average rate at which the paths cross. In order to extend results on the “first-step” probability of intersection and apply the law of large numbers we would need the following assumptions:

• The positions of the two agents are uniformly distributed at every time step (which is the case on the sphere but not on the plane because of reflection problems at the boundary of R), • The crossing-path events are independent over time (which is the case in neither setting because of the strong spatial dependence at consecutive timesteps). Numerous simulations have shown that despite marked departures from these assumptions the long-run rate at which the paths cross is also close to 2d

Section 2 contains the results both for the plane and the sphere. Section 3 is devoted to the numerical simulations. Extensions are discussed in Section 4. Three technical appendices can be found in Section 5.

2.1. Geometric description in the plane and on the sphere. A bounded region of the plane is the most natural setting for agents moving in a 2D environment. However we then need to specify how agents are reflected when they hit the boundary of the region. There is no such problem on a sphere. In what follows R is either a bounded region of the plane or a sphere.

The initial positions V 1 (0) and V 2 (0) of the two agents are assumed uniformly distributed in R. At each time-step the two agents move distances d 1 and d 2 (which are fixed positive parameters) in a straight line (or along a geodesic on a sphere). They depart at random angles α 1 and α 2 that are uniformly and independently distributed over (0, 2π). The endpoints after the k -th time-step are V 1 (k) and V 2 (k) (Figure 1.1).

The definition of a meeting in such a model is tricky because the probability of the two agents being in exactly the same position at any given period is 0. There are however different ways of approximating such a meeting.

One could say that the agents meet if the distance between two points V 1 (k) and V 2 (k) is less than some ǫ. In such a definition results would depend on ǫ, which is undesirable. For this reason we choose to define a meeting during the k -th timestep when paths cross between k and k + 1. This means that the segments (or the “geodesic arcs”)

and V 2 (k) can then be close without the paths crossing, but at least the definition does not depend on some arbitrary ǫ.

The point m 1 in Figure 1.1 is an example of such a synchronous crossing of paths. Of course the agents are not at m 1 at the same time. In the figure the paths cross asynchronously at m 2 .

Much will depend on whether V 1 (k) and V 2 (k) are uniformly distributed on R for every k. This will be the case if R is a sphere because V 1 (0) and V 2 (0) are themselves uniformly distributed. If on the other hand R is a bounded region of the plane, then the uniformity in the distributions of V 1 (k)

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