Phase transitions of semi-scale invariant random fractals

Speaker: Erik Broman, Senior Lecturer, Chalmers/University of Gothenburg, Sweden      

Abstract: 

In all semi-scale invariant 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 (see for example the classical papers

by Dvoretsky or Shepp).

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. 

The talk will be non-technical and is aimed at a broad audience.