Topics in spatial and dynamical phase transitions of interacting particle systems

Phase transition;Approximation algorithm;Gibbs measures;Reconstruction;Constraint satisfaction problem;Glauber dynamics;Spatial mixing;Lattice gas;Extremality of Gibbs measures;Uniqueness of Gibbs measures;Coloring;Random graphs

Restrepo Lopez, Ricardo ; Mathematics

University:Georgia Institute of Technology

Department:Mathematics

关键词: Phase transition;    Approximation algorithm;    Gibbs measures;    Reconstruction;    Constraint satisfaction problem;    Glauber dynamics;    Spatial mixing;    Lattice gas;    Extremality of Gibbs measures;    Uniqueness of Gibbs measures;    Coloring;    Random graphs;   

Others  :  https://smartech.gatech.edu/bitstream/1853/42729/1/restrepolopez_ricardo_201112_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

In this work we provide several improvements in the study of phase transitionsof 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 ofthe 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 phasetransition for the mixing time of the Glauber dynamics at the same location ofthe 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.