Onsdag 20 september 2017, Erik Broman, Chalmers tekniska högskola
Talare: Erik Broman, Chalmers tekniska högskola
Titel: Covering a subset of R^d by Poissonian random sets
Tid och plats: Onsdag 20 september 2017, Hopningspunkten, 15.15-16.15
Sammanfattning: The problem of covering a set A by a collection of random sets dates back to Dvoretzky in 1954. Since then, a host of papers have been written on the subject. In this talk we shall review some of this history and discuss two directions in which progress have recently been made.
In the first case we consider a statistically scale invariant collection of subsets of R^d, which are chosen at random according to a Poisson process of intensity lambda. The complement of the union of these sets is then a random fractal that we denote by C. Such random fractals have been studied in many contexts, but here we are interested in the critical value of lambda for which the set C is almost surely empty (so that R^d is completely covered). Such problems were earlier studied and solved in one dimension, while here we shall present recent progress which solves it in all dimensions. This part is based on joint work with J. Jonasson and J. Tykesson.
In the second direction we consider a dynamic version of coverings. For instance, the set A could be a box of side lengths n, and then balls are raining from the sky at unit rate. One then asks for the time at which A is covered. Together with F. Mussini I have recently studied a variant in which the balls are replaced by bi-infinite cylinders. This makes the problem fundamentally different as one no longer have independence between well separated regions. Thus, new methods and techniques must be used. Our main result is that we find the correct asymptotics for the cover time as the set A grows.
Senast uppdaterad: Tue Sep 05 11:48:33 CEST 2017