We study the mixing time of a Markov chain M nn on biased permutations, a problem arising in the context of self-organizing lists. In each step, M nn chooses two adjacent elements k, and and exchanges their positions with probability p ,k . Here we define two general classes and give the first proofs that the chain is rapidly mixing for both. We also demonstrate that the chain is not always rapidly mixing.

The Schelling segregation model attempts to explain possible causes of racial segregation in cities. Schelling considered residents of two types, where everyone prefers that the majority of his or her neighbors are of the same type. He showed through simulations that even mild preferences of this type can lead to segregation if residents move whenever they are not happy with their local environments. We generalize the Schelling model to include a broad class of bias functions determining individuals happiness or desire to move, called the General Influence Model. We show that for any influence function in this class, the dynamics will be rapidly mixing and cities will be integrated (i.e., there will not be clustering) if the racial bias is sufficiently low. Next we show complementary results for two broad classes of influence functions: Increasing Bias Functions (IBF), where an individual's likelihood of moving increases each time someone of the same color leaves (this does not include Schelling's threshold models), and Threshold Bias Functions (TBF) with the threshold exceeding one half, reminiscent of the model Schelling originally proposed. For both classes (IBF and TBF), we show that when the bias is sufficiently high, the dynamics take exponential time to mix and we will have segregation and a large "ghetto" will form.

In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements i and j are placed in order (i, j) with probability p i,j . Our goal is to identify the class of parameter sets P = {p i,j } for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill that all monotone, positively biased distributions are rapidly mixing.We resolve Fill's conjecture in the affirmative for distributions arising from k-class particle processes, where the elements are divided into k classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that k is a constant, and all probabilities between elements in different classes are bounded away from 1/2. These particle processes arise in the context of self-organizing lists and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Additionally we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et. al. (SODA '13). Our work generalizes recent work by Haddadan and Winkler (STACS '17) studying 3-class particle processes.Our proof involves analyzing a generalized biased exclusion process, which is a nearestneighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. (SODA '09) and Benjamini et. al (Trans. AMS '05) on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.The fundamental problem of generating a random permutation has a long history in computer science, beginning as early as 1969 [13]. One way to generate a random permutation is to use the nearest-neighbor Markov chain M nn which repeatedly swaps the elements in a random pair of adjacent positions. The chain M nn was among the first considered in the study of the computational efficiency of Markov chains for sampling [1,6,4] and has subsequently been studied extensively. After a series of papers, the mixing time of M nn (Θ(n 3 log n) [25]) is now well-understood when sampling from the uniform distribution on the permutation group S n .The nearest-neighbor chain M nn can also be used to sample from more general probability distributions by allowing non-uniform swap probabilities. Suppose we have a set of parameters P = {p i,j } and that M nn puts neighboring elements i and j in order (i, j) with probability p i,j , where p j,i = 1 − p i,j . Despite the simplicity of this natural extension, much less is known about the mixing time of M nn in the non-uniform case. In this paper we look at the question for which parameter sets P is M nn rapidly (polynomially) mixing? We say P is positively

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.

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