Starting from any graph on {1, … , n}, consider the Markov chain where at each time-step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of nonempty forests on {1, … , n}. The random forest corresponding to this stationary distribution has interesting connections with the uniform rooted labeled tree and the uniform attachment tree. We fully characterize its degree distribution, the distribution of its number of trees, and the limit distribution of the size of a tree sampled uniformly. We also show that the size of the largest tree is asymptotically , where , and that the degree of the most connected vertex is asymptotically .

中文翻译：

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莫兰森林

从 {1, … , n }上的任何图开始 ，考虑马尔可夫链，其中在每个时间步，一个统一选择的顶点与其所有邻居断开连接，并重新连接到另一个统一选择的顶点。该马尔可夫链具有平稳分布，其支持度是 {1, … , n }上的非空森林集 。对应于这种平稳分布的随机森林与统一有根标记树和统一附件树有有趣的联系。我们充分刻画了它的度分布，它的树数分布，以及均匀采样的树的大小极限分布。我们还表明最大树的大小是渐近的，其中，并且最连接的顶点的度数是渐近的。