Convergence rates of Markov chains for some self-assembly and non-saturated Ising models

Algorithms based on Markov chains are ubiquitous across scientific disciplines as they provide a method for extracting statistical information about large, complicated systems. For some self-assembly models, Markov chains can be used to predict both equilibrium and non-equilibrium dynamics. In fact, the efficiency of these self-assembly algorithms can be related to the rate at which simple chains converge to their stationary distribution. We give an overview of the theory of Markov chains and show how many natural chains, including some relevant in the context of self-assembly, undergo a phase transition as a parameter of the model representing temperature is varied. We illustrate this behavior for the non-saturated Ising model in which there are two types of tiles that prefer to be next to other tiles of the same type. Unlike the standard Ising model, we also allow empty spaces that are not occupied by either type of tile. We prove that a local Markov chain that allows tiles to attach and detach from the lattice, the rate of convergence is fast at high temperature and slow at low temperature.