Sampling from Gibbs Distributions
Abstract: This thesis considers computational questions concerning statistical mechanical models of idealized physical systems. The equilibrium state of the physical system is described by a probability distribution over the allowed configurations, known as a Gibbs distribution. By sampling at random from the Gibbs distribution one can study essentially all the thermodynamic properties of the system.
The standard approach to sampling from the Gibbs distribution is the "Markov Chain Monte Carlo" method. At the heart of this method is a Markov chain whose stationary distribution is the Gibbs distribution and which quickly converges to stationarity. For a wide class of physical systems, there is a class of very simple Markov chains known as the "Glauber dynamics," whose moves correspond to local perturbations of the current configuration. These chains are of interest because they are simple, natural and widely used in practice.
We study the properties of the Glauber dynamics in two models of particular combinatorial interest: the Potts model, whose configurations are the set of proper colorings of a graph; and the hard core model, whose configurations are the set of independent sets of a graph. For a range of parameter values we prove that the Glauber dynamics in these models quickly converges to the Gibbs distribution, while in another range of values the time to reach the stationary distribution grows exponentially with the volume of the system. Our results also address the conjectured connection between the convergence rate of the Glauber dynamics and phase transitions in the macroscopic properties of the system.