When agents cross paths?

[Paul W. Box]

Hello group

Here is a question that I got thinking about today.

Imagine that you have a 2D (or anything-D) surface with things (agents)
running around on it in a random fashion, such as the simpleSwarmBugs in
the tutorial. Does anyone know of a way to find the expectation that

a) a particular point (cell) on a discretized surface would be trodden
upon more than once, 

b) a particular point (cell) on a discretized surface would be trodden
upon by one or more individuals, and

c) any two paths cross each other (discrete or continuous surface).

These would each obviously be a function of the density of individuals
in the space and the amount of time that the simulation runs.  I am not
sure of what other factors would be relevant yet.

Has anyone on this list had to deal with a formal description of this? 
Could anyone point me to an article or textbook that discusses anything
similar?

Any ideas would be appreciated.

-- 
/**********************************************************************/
/*  Paul Box                      |        Cogito ergo                */
/*  Dept. of Geography            |      Oculum Dioscoreum            */
/*  University of Florida         |                                   */
/*  address@hidden   |     (I think, therefor            */
/*  http://grove.ufl.edu/~sanduku |           eye yam)                */
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