Clustering in Interfering Binary Mixtures

Abstract

Abstract. Colloids 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. We characterize the high and low density phases for a general family of discrete 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 to perimeter ratio and very high density of one type of molecule. A special case is mixtures of squares and diamonds on Z 2 which corresond to the Ising model at fixed magnetization.

Clustering and Mixing Times for Segregation Models on ℤ2

“…Our proofs build on combinatorial insights developed in Section 3.1 and in [17] to establish clustering (i.e., segregation) for the IBF and TBF models when the bias is high. We characterize clustering by the existence of a region R that has large (quadratic) area, small (linear) perimeter, and whose interior is dense with one of the two colors.…”

“…In order to characterize whether a configuration is segregated or integrated, we determine whether one group of residents has "clustered." We build on a concept of clustering developed in [17] based on the presence of a large region with small perimeter that is densely filled with either R-or B-faces. In Section 4, we will show that a random sample from our model will be exponentially likely to be clustered when the bias is high, and exponentially unlikely to be clustered when the bias is sufficiently low.…”

“…We characterize clustering by the existence of a region R that has large (quadratic) area, small (linear) perimeter, and whose interior is dense with one of the two colors. A similar notion of clustering was used in [17], but the proofs required the introduction of r-bridges and fat contours to handle unoccupied houses and large radii of influence.…”

“…Specifically, we prove that at high values of λ, a typical configuration will have no large contours and will have high density of either Ror B-faces. We combine techniques used to show clustering [17] with the slow mixing techniques used in Section 3.1. Let ρ R be the density of R-faces and ρ B be the density of B-faces.…”

Cycle Basis Markov Chains for the Ising Model

“…The remaining subgraphs are in bijection with spin configurations and inherit the bottleneck of M GD . However, as these bases are nonlocal, the argument requires careful examination of the stationary distribution and we appeal to results of [18] to upper-bound the probability that a spin configuration has a density of +1 spins that is close to 1/2.…”