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  1. Abstract

    We study the mixing time of a Markov chain on biased permutations, a problem related to self‐organizing lists. We are given probabilities for all such that . The chain iteratively chooses two adjacent elements and , and swaps them with probability . It has been conjectured that is rapidly mixing whenever the set of probabilities are “positively biased,” that is, for all . We define two general classes and give the first proofs that is rapidly mixing for both. We also demonstrate that the chain can have exponential mixing time, disproving the most general version of this conjecture.

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  2. Abstract Monotonic surfaces spanning finite regions of ℤ d arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that iteratively adds or removes a single cube below the surface during a step. We consider a biased version of the chain, where we are more likely to add a cube than to remove it, thereby favouring surfaces that are ‘higher’ or have more cubes below it. We prove that the chain is rapidly mixing for any uniform bias in ℤ 2 and for bias λ > d in ℤ d when d  > 2. In ℤ 2 we match the optimal mixing time achieved by Benjamini, Berger, Hoffman and Mossel in the context of biased card shuffling [2], but using much simpler arguments. The proofs use 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. We show that the chain continues to be rapidly mixing if the biases are close to uniform, but that the chain can converge exponentially slowly in the general setting. 
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  3. Groups of interacting active particles, insects, or humans can form clusters that hinder the goals of the collective; therefore, development of robust strategies for control of such clogs is essential, particularly in confined environments. Our biological and robophysical excavation experiments, supported by computational and theoretical models, reveal that digging performance can be robustly optimized within the constraints of narrow tunnels by individual idleness and retreating. Tools from the study of dense particulate ensembles elucidate how idleness reduces the frequency of flow-stopping clogs and how selective retreating reduces cluster dissolution time for the rare clusters that still occur. Our results point to strategies by which dense active matter and swarms can become task capable without sophisticated sensing, planning, and global control of the collective. 
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