Topics in spatial and dynamical phase transitions of interacting particle systems

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Title: Topics in spatial and dynamical phase transitions of interacting particle systems
Author: Restrepo Lopez, Ricardo
Abstract: In this work we provide several improvements in the study of phase transitions of interacting particle systems: - 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 on 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. - 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 analogous hard-hexagon in 1980. - 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 a model; thus providing further evidence for the conjectural algorithmic 'hardness' occurring at such a point.
Type: Dissertation
Date: 2011-08-19
Publisher: Georgia Institute of Technology
Subject: Phase transition
Approximation algorithm
Gibbs measures
Constraint satisfaction problem
Glauber dynamics
Spatial mixing
Lattice gas
Extremality of Gibbs measures
Uniqueness of Gibbs measures
Random graphs
Phase transformations (Statistical physics)
Interfaces (Physical sciences)
Approximation algorithms
Department: Mathematics
Advisor: Committee Chair: Prasad Tetali; Committee Member: Eric Vigoda; Committee Member: Luca Dieci; Committee Member: Ton Dieker; Committee Member: Yuri Bakhtin
Degree: PhD

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