线性扩张上提升马尔可夫链的性质
Properties of the Promotion Markov Chain on Linear Extensions
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，并且特征值在转移概率中是线性的。这一结果已被不同的人以不同的方式加以推广。在这项工作中，我们研究了由Ayyer，Klee和Schilling在2014年引入的偏序集$P$的线性扩展上的扩展推广马尔可夫链给出的推广之一。他们证明，如果偏序集$P$是一个有根森林，这个马尔可夫链的转移阵的特征值在转移概率中是线性的，并描述了它们的多重性。我们表明，同样的属性适用于更大的一类偏序集，我们也得到收敛平稳性的结果。
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. This result has been generalized in different ways by various people. In this work we investigate one of the generalizations given by the extended promotion Markov Chain on linear extensions of a poset $P$ introduced by Ayyer, Klee, and Schilling in 2014. They showed that if the poset $P$ is a rooted forest, the transition matrix of this Markov chain has eigenvalues that are linear in the transition probabilities and described their multiplicities. We show that the same property holds for a larger class of posets for which we also derive convergence to stationarity results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
