MARK KAC SEMINAR

Rényi's random parking process on a domain D in d-space is a point process with hard-core and no-empty-space properties that are desirable for modelling materials such as rubber. It is obtained as follows: particles arrive sequentially at uniform random locations in D, and are rejected if they violate the hard-core constraint, until the accepted particles saturate D.

We describe how any real-valued functional on this point process, provided it enjoys certain subadditivity properties, satisfies an averaging property in the thermodynamic limit. Consequently in this limit, one has a convergence of macroscopically-defined energy functionals for deformations of the point process, to a homogenized limiting energy functional. We may also apply the results to derive laws of large numbers for classical optimization problems such as travelling salesman on the parking point process.

This is joint work with Antoine Gloria.

Run a Brownian motion on a torus for a long time. How large are the random gaps left behind when the path is removed?

In three (or more) dimensions, we find that there is a deterministic spatial scale common to all the large gaps anywhere in the torus. Moreover, we can identify whether a gap of a given shape is likely to exist on this scale, in terms of a single parameter, the classical (Newtonian) capacity. I will describe why this allows us to identify a well-defined “component” structure in our random porous set.

Based on joint work with Frank den Hollander.