Mixing Times for Diffusive Lattice-Based Markov Chains

Graduation date: 2018Markov chains have long been used to sample from probability distributions and simulate dynamical systems. In both cases we would like to know how long it takes for the chain's distribution to converge to within $\varepsilon$ of the stationary distribution in total variation distance; the answer to this is $t_{\rm mix}(\varepsilon)$, called the mixing time of the chain. After this time, we can sample the Markov chain's position to approximate sampling the underlying stationary distribution, and the chain's dynamics exhibit equilibrium behavior. In this dissertation we study the effect that the system size (diameter of the state space, say) has on the mixing time of a sequence $\left( x^{(n)}, \mathcal{L}^{(n)}, \mathcal{P}^{(n)} \right)$ of Markov chains which have locally-finite lattice state spaces, and transition probability functions converging to those of a gradient dynamical system. In lieu of a precise functional form for the mixing time, the problem we study is the asymptotic growth rate as $n \to \infty$.\ud \ud This dissertation offers a novel solution utilizing the weak scaling limits of the Markov chains and local central limit theory for random walks on lattices. By separating the state space into high-drift and low-drift regions where weak limits are valid, and utilizing martingale theory to approximate the chain's behavior in the intermediate regions, we determine the time required for the distribution to converge weakly to to stationarity. This decomposition of the state space makes evaluating the existence of cutoff straightforward as well. Then a local limit theorem is used to strengthen the weak convergence to total variation convergence. The theory developed is then applied to recover recent mixing time results for statistical mechanical models, giving independent proof of known mixing behavior in the mean-field Ising and Potts models, as well as a full description of the mixing behavior of the Blume-Capel model