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<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>

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This thesis presents a statistical mechanical analysis of different formulations of quasispecies theory of molecular evolution. These theories, characterized by two different families of models, the Crow-Kimura and the Eigen model, constitute a microscopie description of evolution. These models are most often used for RNA viruses, where a phase transition is predicted, in agreement with experiments, between an organized or quasispecies phase, and a disordered non-selective phase when the mutation rate exceeds a critical value. The methods of statistical mechanics, in particular field-theoretic methods, are employed to obtain analytic solutions to four problems relevant to biological interest. The first chapter presents the study of evolution under a multiple-peak fitness landscape, with biological applications in the study of the proliferation of viruses or cancer under the control of drugs or the immune system. The second chapter studies the effect of incorporating different forms of horizontal gene transfer and two-parent recombination to the classical formulation of quasispecies models. As an example, we study the effect of the sign of epistasis of the fitness landscape on the advantage or disadvantage of recombination for the mean fitness. The third chapter considers the relaxation of the purine/pyrimidine assumption in the classical formulation of the models, by formulating and solving the parallel and Eigen models in the context of a four-letter alphabet. The fourth and final chapter studies finite population effects, both in the presence and in the absence of horizontal gene transfer.

Green's operator for Hamiltonians with Coulomb plus polynomial potentials