### Topics in Spatial and Dynamical Phase Transitions of Interacting Particle Systems

- Series
- Dissertation Defense
- Time
- Monday, August 15, 2011 - 11:00 for 2 hours
- Location
- Skiles 005
- Speaker
- Ricardo Restrepo Lopez – School of Mathematics, Georgia Tech

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.