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Topics in spatial and dynamical phase transitions of interacting particle systems

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.

Identiferoai:union.ndltd.org:GATECH/oai:smartech.gatech.edu:1853/42729
Date19 August 2011
CreatorsRestrepo Lopez, Ricardo
PublisherGeorgia Institute of Technology
Source SetsGeorgia Tech Electronic Thesis and Dissertation Archive
Detected LanguageEnglish
TypeDissertation

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