Proefschrift

# Dynamical Gibbs-non-Gibbs transitions and Brownian percolation

Promotor: Prof.dr. W.Th.F. den Hollander, Co-promotor: R. Fernandez.

 Auteur J.M. Martinez Datum 25 maart 2014 Links Thesis in Leiden Repository

This thesis deals with two different models in two different contexts. The first part deals with dynamical Gibbs-non-Gibbs transitions. Gibbs measures describe the equilibrium states of a system consisting of a large number of components that interact with each other. Due to the large number of particles, it is natural to assume that the state of the system is random. Gibbs measures capture this randomness. This description involves some particular regularity'' conditions for the conditional probabilities. The question of interest is whether this condition remains valid after the system is subjected to a stochastic dynamic. Is it still possible to describe the evolved measure as a Gibbs measure? The second part deals with stochastic geometry. The relevant information about the particles is their position. Particles may be placed at random in any region of the space. Subsequently, each particle is displaced independently of each other according to a d-dimensional Brownian Motion during t time, and the trace produced by that motion is recorded. The question of interest is whether the final set obtained from all the traces has an infinite connected component or not. If so, then is it unique?