## Markov chains at the interface of combinatorics, computing, and statistical physics

##### Abstract

The fields of statistical physics, discrete probability, combinatorics, and theoretical computer science have converged around efforts to understand random structures and algorithms. Recent activity in the interface of these fields has enabled tremendous breakthroughs in each domain and has supplied a new set of techniques for researchers approaching related problems. This thesis makes progress on several problems in this interface whose solutions all build on insights from multiple disciplinary perspectives.
First, we consider a dynamic growth process arising in the context of DNA-based self-assembly. The assembly process can be modeled as a simple Markov chain. We prove that the chain is rapidly mixing for large enough bias in regions of Z^d. The proof uses a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile, which arises in the nanotechnology application. Moreover, we use intuition from statistical physics to construct a choice of the biases for which the Markov chain M_mon requires exponential time to converge.
Second, we consider a related problem regarding the convergence rate of biased permutations that arises in the context of self-organizing lists. The Markov chain M_nn in this case is a nearest-neighbor chain that allows adjacent transpositions, and the rate of these exchanges is governed by various input parameters. It was conjectured that the chain is
always rapidly mixing when the inversion probabilities are positively biased, i.e., we put nearest neighbor pair x<y
in order with bias 1/2 <= p_{xy} <= 1 and out of order with bias
1-p_{xy}. The Markov chain M_mon was known to have connections to a simplified version of this biased card-shuffling. We provide new connections between M_nn and M_mon by using simple combinatorial bijections, and we prove that M_nn is always rapidly mixing for two general classes of positively biased {p_{xy}}. More significantly, we also prove that the general conjecture is false by exhibiting values for the p_{xy}, with
1/2 <= p_{xy} <= 1 for all x< y, but for which the transposition chain will require exponential time to converge.
Finally, we consider a model of colloids, which are binary mixtures of molecules with one type of molecule suspended in another. It is believed that at low density typical configurations will be well-mixed throughout, while at high density they will separate into clusters. This clustering has proved elusive to verify, since all local sampling algorithms are known to be inefficient at high density, and in fact a new nonlocal algorithm was recently shown to require exponential time in some cases.
We characterize the high and low density phases for a general family of discrete {it interfering binary mixtures} by showing that they exhibit a "clustering property' at high density and not at low density. The clustering property states that
there will be a region that has very high area, very small perimeter, and high density of one type of molecule. Special cases of interfering binary mixtures include the Ising model at fixed magnetization and independent sets.