$k$类有偏置换的快速混合
Rapid Mixing of $k$-Class Biased Permutations
摘要：
本文研究了排列集合上最近邻换位马氏链的一个有偏版本，其中相邻元素$i$和$j$以概率$p_{i,j}$按$(i,j)$顺序排列.我们的目标是确定类的参数集${\bf P} = \{p_{i,j}\}$的马尔可夫链是快速混合。具体来说，我们认为开放的猜想吉姆填充，所有单调，正偏态分布迅速混合。我们解决填充的猜想肯定的分布所产生的$k$类粒子过程，其中的元素被分为$k$类和交换相邻元素的概率取决于特定的类的元素。我们进一步要求$k$是一个常数，并且不同类中元素之间的所有概率都有界远离$1/2$。这些粒子过程出现在自组织列表的上下文中，我们的结果也适用于排列之外的设置，其中一类中的所有粒子是不可区分的。此外，我们表明，更广泛的一类分布的基础上的树也迅速混合，概括了一类分析Bhakta等。（SODA '13）。我们的工作概括了Haddadan和Winkler（STACS '17）最近研究3类粒子过程的工作。我们的证明涉及分析一个广义有偏排斥过程，这是一个最近邻换位链应用到一个2粒子系统。有偏排斥过程是独立的兴趣，在自组装中的应用。我们推广了Greenberg等人（SODA '09）和Benjamini等人的结果。al（trans.AMS '05）关于偏置排除过程，以允许交换相邻元素的概率取决于整个系统，只要最小偏置远离$1$。
英文摘要：
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 ${\bf P} = \{p_{i,j}\}$ 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 nearest-neighbor 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 $1$.
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