使用扩展提升的树上随机洗牌
Random shuffles on trees using extended promotion
摘要：
Tsetlin图书馆是一个很好的研究模型，可以用来描述图书馆书架上书籍的排列方式随着时间的推移而演变。这个马尔可夫链最有趣的性质之一是它的谱可以精确计算，并且本征值在转移概率中是线性的。在本文中，我们考虑了一个推广，它可以被解释为一个自组织图书馆，其中的书籍在每个书架上的安排被限制为一个固定偏序集的线性扩展。书籍的移动由Ayyer、Klee和Schilling的扩展促销操作员给出，而书架、书架等则根据移动到后面的移动而发展，就像Björner的自组织图书馆一样。我们表明，这个马尔可夫链的转移阵的特征值是$\pm 1$的转移概率的整数组合，如果规定的书安排的限制的偏序集是根森林或更一般地说，如果它们由一个根森林和所谓的梯子的顺序和。对于一些结果，我们表明，由移动所产生的幺半群是$\mathcal{R}$ -平凡的，或更一般地说，在$\textbf{DO(Ab)}$，然后我们使用的理论左随机游动的极小理想，这样的幺半群找到的特征值。此外，为了给一个组合描述的特征值在更一般的情况下，我们涉及的特征值时，限制的书籍安排的变化，只允许一个额外的换位两个固定的书籍。
英文摘要：
The Tsetlin library is a very well studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. In this paper we consider a generalization which can be interpreted as a self-organizing library in which the arrangements of books on each shelf are restricted to be linear extensions of a fixed poset. The moves on the books are given by the extended promotion operators of Ayyer, Klee, and Schilling while the shelves, bookcases, etc. evolve according to the move-to-back moves as in the the self-organizing library of Björner. We show that the eigenvalues of the transition matrix of this Markov chain are $\pm 1$ integer combinations of the transition probabilities if the posets that prescribe the restrictions on the book arrangements are rooted forests or more generally, if they consist of ordinal sums of a rooted forest and so called ladders. For some of the results we show that the monoids generated by the moves are either $\mathcal{R}$-trivial or, more generally, in $\textbf{DO(Ab)}$ and then we use the theory of left random walks on the minimal ideal of such monoids to find the eigenvalues. Moreover, in order to give a combinatorial description of the eigenvalues in the more general case, we relate the eigenvalues when the restrictions on the book arrangements change only by allowing for one additional transposition of two fixed books.
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