Percolation in correlated systems

In this thesis we study various problems in dependent percolation theory. In the first part of this thesis we study disordered q-state Potts models as examples of systems in which there is percolation for an arbitrary low density and no percolation for arbitrary high density of occupied sites. In the second part of the thesis we study dependent percolation models in which the correlations between the site occupation variables are long range, i.e. decaying as r [superscript -a] for a [less than sign] d, where r is the separation between any two sites and d is the dimension of the model. Scaling analysis suggests that such long range correlated percolation models define a new percolation universality classes with critical exponents depending on a. We develop a field theoretic description of these models in an attempt to calculate the critical exponents of the transition in an double expansion in terms of epsilon = 6 - d and delta = 4 - a. In the third part we study the percolation transition for two specific long range correlated percolation models on the 3 dimensional integer square lattice. These two percolation models are derived from the Voter model and the Harmonic crystal respectively. Our simulation results confirm the basic scaling arguments and the field theoretic results.Ph.D.Includes bibliographical references (p. 71-74)