Topics in percolation, polymers and Potts dynamics

In this thesis we investigate different aspects of various well-known models in statistical mechanics as well as one model which is relatively less standard. There are essentially four (rather disjoint) topics/problems which are treated in this work:;Finite Connections for Supercritical Bernoulli Bond Percolation in 2D. Two vertices x and y are said to be finitely connected if they belong to the same cluster and this cluster is finite. We derive sharp asymptotics of finite connections for super-critical Bernoulli bond percolation on Z2 .;Directed Polymers in Random Environment with Heavy Tails. We study the model of Directed Polymers in Random Environment in 1..1 dimensions, where the distribution at a site has a tail which decays regularly polynomially with power alpha, where alpha ∈ (0, 2). After proper scaling of temperature beta -1, we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (alpha, beta)-indexed family of measures on 1-Lipschitz curves on [0, 1].;Glauber Dynamics for the Curie-Weiss Potts Model: Mixing Time Analysis. Here, we consider the discrete-time Glauber dynamics for the q-states Potts model (q ⩾ 3) on the complete graph with n vertices and analyze its mixing time, namely the time it takes until the state distribution is epsilon-close to the Potts distribution, starting from the worst possible initial state. We show that there exists a critical inverse temperature beta d(q) ∈ (0, betac( q)) above which mixing time is exponential and below which mixing time is asymptotically C(beta, q) n log(n) (here betac( q) stands for the order/disorder threshold). At criticality mixing time is theta(n4/3). beta d(q) can also be characterized as the largest temperature at which the system exhibits meta-stability.;Fixation for Distributed Clustering Processes. Lastly, we study a discrete-time resource flow in Zd , where wealthier vertices attract the resources of their less rich neighbors. For any translation-invariant probability distribution of initial resource quantities, we prove that the flow at each vertex terminates after finitely many steps.