无向图空间上不可约马尔可夫链

离散环面3-着色上局部马尔可夫链的Torpid混合

Torpid Mixing of Local Markov Chains on 3-Colorings of the Discrete Torus

本文研究离散环面$T_{L,d}={0,..., L-1}^d$的3-着色抽样的局部马尔可夫链。证明了存在一个常数$ρ\approx .22$，使得对于所有足够大的偶数$L \geq 4$和$d$，某些局部马尔可夫链需要指数时间才能收敛到平衡点.更准确地说，如果$\cM$是$T_{L,d}$的真3-着色集上的马尔可夫链，在每一步更新至多$ρL^d$顶点的颜色，并且其平稳分布是均匀的，则$\cM$在$L^{d-1}$中收敛到平稳性是指数的。我们的证明是基于电导参数，建立在敏感的新组合枚举技术。

A Markov chain on the solution space of edge-colorings of bipartite graphs

In this paper, we exhibit an irreducible Markov chain $M$ on the edge $k$-colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis-Hastings algorithm when the algorithm is applied on $M$ with the uniform distribution of all possible edge $k$-colorings of $G$. A special case of our results is the solution space of the possible completions of Latin rectangles.

Switch-based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on $n$ vertices. As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least $n/2+7$ can be transformed into each other by switch operations of size at most $10$, implying that the switch Markov chain using switches of size at most $10$ is irreducible. As a proof of concept, we also show that this Markov chain is rapidly mixing on dense monotone graphs.

采样接近最大度数的颜色配置：非马尔可夫耦合与局部均匀性

Sampling Colorings Close to the Maximum Degree: Non-Markovian Coupling and Local Uniformity

本文研究了在图的最大度数Δ下，采样k-着色问题，证明当k≥(1+δ)Δ时，Glauber动力学在图的圈长≥11且最大度数Δ的情况下具有最优混合时间。

三角格中重组马尔可夫链的不可约性

Irreducibility of Recombination Markov Chains in the Triangular Lattice

该研究针对选区划分抽样中重组马尔可夫链的不可约性理论空白，将其转化为图论问题，证明了三角格中任意大三角区域的三选区重组马尔可夫链在严格规模约束下不可约，填补了相关理论研究的空白。

树上边着色随机采样至最大度情形的最优混合

Optimal Mixing for Randomly Sampling Edge Colorings on Trees Down to the Max Degree

本文证明树上边着色的Glauber动力学在q≥Δ+2时具有O(n)最优松弛时间，并为q=Δ+1情形提出更新相邻边对的替代链，同时给出Δ-正则完全树的混合时间上界。