The switch Markov chain has been extensively studied as the most natural Markov chain Monte Carlo approach for sampling graphs with prescribed degree sequences. We show that the switch chain for sampling simple undirected graphs with a given degree sequence is rapidly mixing when the degree sequence is so‐called strongly stable. Strong stability is satisfied by all degree sequences for which the switch chain was known to be rapidly mixing based on Sinclair's multicommodity flow method up until a recent manuscript of Erdős and coworkers in 2019. Our approach relies on an embedding argument, involving a Markov chain defined by Jerrum and Sinclair in 1990. This results in a much shorter proof that unifies (almost) all the rapid mixing results for the switch chain in the literature, and extends them up to sharp characterizations of P‐stable degree sequences. In particular, our work resolves an open problem posed by Greenhill and Sfragara in 2017.

中文翻译：

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快速混合开关马尔可夫链以获得强稳定度数序列

开关马尔可夫链已被广泛研究为最自然的马尔可夫链蒙特卡洛方法，用于对具有指定次数序列的图形进行采样。我们证明，当阶数序列很强时，用于给定阶数序列的简单无向图采样的交换链正在迅速混合。直到开关链都基于辛克莱的多商品流方法迅速混合的所有度数序列才满足强稳定性，直到2019年Erdős和同事的最新手稿为止。我们的方法依赖于嵌入论点，涉及定义的马尔可夫链由Jerrum和Sinclair在1990年提出。这导致了更短的证明，它统一了（几乎）文献中开关链的所有快速混合结果，并将其扩展到对P稳定度序列的清晰刻画。特别是，我们的工作解决了Greenhill和Sfragara在2017年提出的一个开放性问题。