Mixing Times of the Schelling Segregation Model and Biased Permutations

Markov chains are fundamental tools used throughout the sciences and engineering; the design and analysis of Markov Chains has been a focus of theoretical computer science for the last 20 years. A Markov Chain takes a random walk in a large state space Ω, converging to a target stationary distribution π over Ω. The number of steps needed for the random walk to have distribution close to π is the mixing time. For the Markov Chain to be useful, the mixing time should be polynomial in the input to the problem, typically logarithmic in |Ω|. Such a chain is called rapidly mixing. I am interested in applying Markov chain techniques to applied problems, and have been working on two primary problems: the Schelling Segregation Model, a well studied model in economics, and a biased permutation problem arising from the Move Ahead One list update paradigm in caching. In both cases, we leverage intuition gained from studying models in statistical physics.