<p> In this thesis we study continuous time Markov processes whose state space consists of an assignment of +1 or -1 to each vertex <i>x</i> of a graph <i>G.</i> We will consider two processes, σ(<i> t</i>) and σ'(<i>t</i>), having similar update rules. The process σ(<i>t</i>) 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, σ<i><sub>x</sub>(t),</i> by taking the value of a majority of <i>x</i>'s nearest neighbors or else tossing a fair coin in case of a tie. The process σ'(<i> t</i>) starts from an arbitrary initial configuration and evolves according to the same rules as σ(<i>t</i>), except for some vertices which are frozen plus (resp., minus) with density ρ<sup>+</sup> (resp., & ρ<sup>–</sup>) and whose value is not allowed to change. Our results are for when σ(<i>t</i>) evolves on graphs related to homogeneous trees of degree <i>K</i> ≥ 3, such as finite or infinite stacks of such trees, while the process σ'(<i>t</i>) evolves on Z<sup>d</sup>, <i>d</i> ≥ 2. We study the long time behavior of these processes and, in the case of σ'(<i>t</i>), 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(<i>t</i>) reaches fixation to +1 consensus. For σ'(<i> t</i>) we prove that, if ρ<sup>+</sup>>0 and ρ<sup>– </sup> = 0, all vertices end up as fixed plus, while for ρ<sup>+</sup> >0 and ρ<sup>–</sup> very small (compared to ρ<sup> +</sup>), the fixed minus and flippers together do not percolate.</p>
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:3665138 |
| Date | 19 December 2014 |
| Creators | Eckner, Sinziana Maria |
| Publisher | New York University |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | English |
| Type | thesis |
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