Phase transitions of semi-scale invariant random fractals
Speaker: Erik Broman, Senior Lecturer, Chalmers/University of Gothenburg, Sweden
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Speaker:听 听 听
Abstract:听
In all semi-scale invariant random fractal models, there is an听
intensity parameter $\lambda>0$ of the underlying Poisson process which essentially determines听
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听
connected components, to the phase where it is almost surely totally disconnected.听
Another is when the fractal transitions from being totally disconnected to disappearing听
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).
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In the talk we will present results concerning both of these phase transitions. In particular,听
the results include determination of the exact value of the parameter $\lambda$ at which听
the second transition mentioned occurs. Furthermore, we are able to determine the behavior of the听
fractal sets at the critical points of both of these phase transitions.听
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The talk will be non-technical and is aimed at a broad audience.
