Cancelled! PC Seminar: Phase transitions of semi-scale invariant Poissonian random fractals

Abstract: In all semi-scale invariant Poissonian random fractal models, there is an intensity parameter $\lambda>0$ of the underlying Poisson process which essentially determines the nature of the resulting random fractal. As $\lambda$ varies, the models undergo several phase transitions. One is when the fractal set transitions from containing connected components, to the phase where it is almost surely totally disconnected. Another is when the fractal transitions from being totally disconnected to disappearing completely (i.e. it is empty). As we will explain, this is intimately connected to the classical problem of covering a fixed set by other random sets.

In the talk we will present results concerning both of these phase transitions. In particular, the results include determination of the exact value of the parameter $\lambda$ at which the second transition mentioned occurs. Furthermore, we are able to determine the behavior of the fractal sets at the critical points of both of these phase transitions. At the end, we will also briefly discuss an application to a percolation model in hyperbolic space.

This is a seminar in our seminar series on Probability and Combinatorics (PC).