Dissertation/Thesis Abstract

Stochastic Ising Models at Zero Temperature on Various Graphs
by Eckner, Sinziana Maria, Ph.D., New York University, 2014, 69; 3665138
Abstract (Summary)

In this thesis we study continuous time Markov processes whose state space consists of an assignment of +1 or -1 to each vertex x of a graph G. We will consider two processes, σ( t) and σ'(t), having similar update rules. The process σ(t) starts from an initial spin configuration chosen from a Bernoulli product measure with density θ of +1 spins, and updates the spin at each vertex, σx(t), by taking the value of a majority of x's nearest neighbors or else tossing a fair coin in case of a tie. The process σ'( t) starts from an arbitrary initial configuration and evolves according to the same rules as σ(t), except for some vertices which are frozen plus (resp., minus) with density ρ+ (resp., & ρ) and whose value is not allowed to change. Our results are for when σ(t) evolves on graphs related to homogeneous trees of degree K ≥ 3, such as finite or infinite stacks of such trees, while the process σ'(t) evolves on Zd, d ≥ 2. We study the long time behavior of these processes and, in the case of σ'(t), the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). We prove that, if θ is close enough to 1, σ(t) reaches fixation to +1 consensus. For σ'( t) we prove that, if ρ+>0 and ρ = 0, all vertices end up as fixed plus, while for ρ+ >0 and ρ very small (compared to ρ +), the fixed minus and flippers together do not percolate.

Indexing (document details)
Advisor: Newman, Charles M.
School: New York University
Department: Mathematics
School Location: United States -- New York
Source: DAI-B 76/04(E), Dissertation Abstracts International
Subjects: Mathematics, Theoretical physics
Keywords: Coarsening dynamics, Glauber dynamics, Ising model, Probability theory, Statistical mechanics, Statistical physics
Publication Number: 3665138
ISBN: 9781321374414
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