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.
|Advisor:||Newman, Charles M.|
|School:||New York University|
|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|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be