Topics in Spatial and Dynamical Phase Transitions of Interacting Particle Systems

Dissertation Defense
Monday, August 15, 2011 - 11:00am for 2 hours
Skiles 005
Ricardo Restrepo Lopez – School of Mathematics, Georgia Tech
Ricardo Restrepo
In this work we provide several improvements in the study of phase transitions of interacting particle systems: 1. We determine a quantitative relation between non-extremality of the limiting Gibbs measure of a tree-based spin system, and the temporal mixing of the Glauber Dynamics over its finite projections. We define the concept of `sensitivity' of a reconstruction scheme to establish such a relation. In particular, we focus in the independent sets model, determining a phase transition for the mixing time of the Glauber dynamics at the same location of the extremality threshold of the simple invariant Gibbs version of the model. 2. We develop the technical analysis of the so-called spatial mixing conditions for interacting particle systems to account for the connectivity structure of the underlying graph. This analysis leads to improvements regarding the location of the uniqueness/non-uniqueness phase transition for the independent sets model over amenable graphs; among them, the elusive hard-square model in lattice statistics, which has received attention since Baxter's solution of the analogue hard-hexagon in 1980. 3. We build on the work of Montanari and Gerschenfeld to determine the existence of correlations for the coloring model in sparse random graphs. In particular, we prove that correlations exist above the `clustering' threshold of such model; thus providing further evidence for the conjectural algorithmic `hardness' occurring at such point.