发表机构
Simons Institute for the Theory of Computing; University of California Berkeley（西蒙斯计算理论研究所; 加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过高维展开理论分析开关链的混合时间，证明了在边数满足一定条件时混合时间为O(Δ²m log m)，解决了Cooper等人的猜想。
AI 中文摘要
开关链是一个定义在给定图形度序列的标记实现集合上的马尔可夫链。在每一步中，随机选择一对顶点不相交的边，然后过程尝试用同一四个顶点的均匀选择的完美匹配替换它们，拒绝任何会产生多重边的提议。所得过程相对于所有实现上的均匀分布是可逆的。我们通过将实现视为单纯复形的面并研究原始过程的一个变体（称为单纯开关链）来研究该链的混合时间，我们使用高维展开理论中的工具对其进行分析。我们的技术贡献包括证明足够高余维数的面的链环是强谱展开器，以及比较大块更新和双边缘更新的狄利克雷能量。我们的主要结果是当$m\ge C\Delta^{8}$时，单纯和经典开关链的混合时间均为$O(\Delta^{2}m\log m)$，其中$m$是边数，$\Delta$是最大规定度数，$C>0$是一个绝对常数。对于具有固定最大度数的$n$个顶点上的序列，这证明了链在$O(n\log n)$步内混合，解决了Cooper、Dyer和Greenhill的长期猜想，并将其结论扩展到不规则度序列。
英文摘要
The switch chain is a Markov chain defined on the set of labelled realizations of a given graphical degree sequence. At each step, a pair of vertex-disjoint edges is chosen at random and the process attempts to replace them with a uniformly chosen perfect matching of the same four vertices, rejecting any proposal that would create a multiple edge. The resulting process is reversible with respect to the uniform distribution on all realizations. We investigate the mixing time of this chain by viewing realizations as the facets of a simplicial complex and studying a variant of the original process called the simplicial switch chain, which we analyze using tools from the theory of high-dimensional expansion. Our technical contributions include a proof that links of faces of sufficiently high codimension are strong spectral expanders and a comparison between the Dirichlet energies of large block updates and two-edge updates. Our main result is an $O(Δ^{2}m\log m)$ bound on the mixing time of both simplicial and classical switch chains whenever $m\ge CΔ^{8}$, where $m$ is the number of edges, $Δ$ is the maximum prescribed degree, and $C>0$ is an absolute constant. For sequences on $n$ vertices with fixed maximum degree, this proves that the chain mixes in $O(n\log n)$ steps, resolving a longstanding conjecture of Cooper, Dyer, and Greenhill and extending its conclusion to irregular degree sequences.
Comments52 pages, 3 figures
