<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, &sigma;(<i> t</i>) and &sigma;'(<i>t</i>), having similar update rules. The process &sigma;(<i>t</i>) starts from an initial spin configuration chosen from a Bernoulli product measure with density &theta; of +1 spins, and updates the spin at each vertex, &sigma;<i>_x(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 &sigma;'(<i> t</i>) starts from an arbitrary initial configuration and evolves according to the same rules as &sigma;(<i>t</i>), except for some vertices which are frozen plus (resp., minus) with density &rho;⁺ (resp., &amp; &rho;^&ndash;) and whose value is not allowed to change. Our results are for when &sigma;(<i>t</i>) evolves on graphs related to homogeneous trees of degree <i>K</i> &ge; 3, such as finite or infinite stacks of such trees, while the process &sigma;'(<i>t</i>) evolves on Z^d, <i>d</i> &ge; 2. We study the long time behavior of these processes and, in the case of &sigma;'(<i>t</i>), the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). We prove that, if &theta; is close enough to 1, &sigma;(<i>t</i>) reaches fixation to +1 consensus. For &sigma;'(<i> t</i>) we prove that, if &rho;⁺>0 and &rho;^&ndash; = 0, all vertices end up as fixed plus, while for &rho;⁺ >0 and &rho;^&ndash; very small (compared to &rho;⁺), the fixed minus and flippers together do not percolate.</p>

Thesis (Ph.D.)--Brown University, 2006. / (UMI)AAI3227970. Adviser: Georgios Daskalopoulos. Source: Dissertation Abstracts International, Volume: 67-08, Section: B, page: 4466.