Coupling, Concentration and Random Walks in Dynamic Random Environments

Abstract

In this thesis we discuss concentration inequalities, relaxation to equilibrium of stochastic dynamics, and random walks in dynamic random environments. In stochastic systems one is interested in macroscopic and/or asymptotic properties as well as in fluctuations around typical behaviour. But the dependence structure induced by the interaction between the components of the system makes the analysis challenging. In order to overcome this in different settings a variety of methods are employed. Additive functionals of Markov processes play important roles in applications. In order to get exponential and moment estimates for their fluctuations a non-standard martingale approximation is used. The resulting general theorems do not require special properties like reversibility or a spectral gap. What is needed is some control on the expected evolution. That is, the difference of the evolution starting from two adjacent'' configurations has to be controlled. Coupling methods are well suited to do perform this comparison. In concrete examples couplings are used to prove the conditions of the theorems. In statistical mechanics Gibbs measures and Markov random fields play important roles. The Poincare inequality is an important property describing the regularity of the measure. We prove the Poincare inequality via a martingale telescoping argument. To control the individual increments of the martingale we use a coupling method called disagreement percolation. If the clusters of this percolation are sufficiently small we obtain the Poincare inequality. When interacting spin systems and their dynamics have a delicate connection to their ergodic measure(s) one has to take more care. We carefully study the graphical construction of the dynamics to understand how the influence of the measure can be preserved. An assumption is made that one can control how fast the system in equilibrium can compensate for a single spin flip. Under this assumption we obtain relaxation speed estimates for general functions. In attractive spin systems the condition can be reduced to the decay of auto-correlation of the spin at the origin. An application where this is of use is the low-temperature Ising model. Finally we look at random walks in dynamic random environments. Here a time-changing random environment drives the motion of a particle. The goal is to understand under which conditions the macroscopic behaviour of this random walk is like that of a Brownian motion. We use coupling to prove a law of large numbers as well as a functional central limit theorem for the position of the random walk. Only polynomial decay of correlations in time are needed for the environment, and the influence of the environment on the walk can be very general.