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• W2130431319 abstract "This dissertation studies relationships between fast convergence to equilibrium (mixing in time) of natural Markov chain Monte Carlo algorithms for discrete spin systems, and decay of correlations with distance in the corresponding equilibrium distribution (mixing in space). The results fall into four main groups. In the first part we generalize the Dobrushin and Dobrushin-Shlosman conditions for uniqueness of the Gibbs measure (a form of mixing in space) by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the total influence of a site is small. In the second part we critically examine a known sharp equivalence between appropriate notions of mixing in time and in space. For this part, the discussion applies only to systems on the d-dimensional integer lattice Zd . We give new, purely combinatorial arguments to prove that, if the mixing time of the Glauber dynamics is O(n log n), then spin correlations decay exponentially fast with distance in the Gibbs distribution. In the third part we develop a new framework for analyzing the mixing time for spin systems on trees. The main technical result here is that on trees, an appropriate form of mixing in space implies O( n log n) mixing time of the Glauber dynamics. The novelty of this implication is that it is specific to the boundary condition. This allows us to give the first comprehensive analysis (in any context) of the effect of boundary conditions on the mixing time for the Ising and other models. In the fourth part we explore directions for extending our results for trees to the 2-dimensional integer lattice Z2 . The main motivation here is resolving a long standing conjecture which states that, conditioned on the all-(+) boundary, the mixing time re mains bounded by a fixed polynomial in n at all temperatures. (Notice that for the free boundary case, the mixing time at low temperatures is known to be very slow, specifically exp(Θ( n ))). (Abstract shortened by UMI.)" @default.
• W2130431319 created "2016-06-24" @default.
• W2130431319 creator A5020315523 @default.
• W2130431319 creator A5076599508 @default.
• W2130431319 date "2004-01-01" @default.
• W2130431319 modified "2023-09-26" @default.
• W2130431319 title "Mixing in time and space for discrete spin systems" @default.
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