Systematic Scan Dynamics on the Ising Model

Abstract

The Ising model is one of the most extensively studied particle system models thatoriginated from Statistical physics. With the help of the Gibbs measure and the language of the Markov chain, several questions regarding the mixing times and cutoff phenomena have been dealt with. The Ising type models vary in terms of the underlying graph and the dynamics of how to evolve the given model. Similar to the general card shuffling problems, we can imagine a particle system in which each particle updates in a certain order but with the same rule as the original dynamics. This systematic scan dynamics pose another question, which is often challenging due to the lack of symmetry. We apply the systematic scan idea to the Ising type model. We study the mixing time and the existence of cutoffs of the systematic scan Glauber dynamics Ising model on the complete graph. There is a cutoff in the high-temperature regime, while there is not in the critical and the low-temperature regime. We provide not only the mixing times for each regime but also where exactly the cutoff happens and the window size for the high-temperature regime, in the last chapter of this dissertation. The upper bound can be achieved from the coupling argument, and the lower bound comes by computing the drift of the magnetization.