Rapid mixing of subset Glauber dynamics on graphs of bounded tree-width

Rapid mixing of Glauber dynamics for colorings below Vigoda's $11/6$ threshold

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of $k$-colorings of a graph $G$ on $n$ vertices with maximum degree $Δ$ is rapidly mixing for $k \geq Δ+2$. In FOCS 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper $k$-colorings for $k > \frac{11}{6}Δ$, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the $\frac{11}{6}Δ$ barrier for general graphs by showing rapid mixing for $k > (\frac{11}{6} - η)Δ$ for some positive constant $η$. The key to…

Mixing of the Glauber dynamics for the ferromagnetic Potts model

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree ($Δ$) of the underlying graph and the number of colours or spins ($q$) in determining whether the dynamics mixes rapidly or not. We find a lower bound $L$ on the number of colours such that Glauber dynamics is rapidly mixing if at least $L$ colours are used. We give a closely-matching upper bound $U$ on the number of colours such that with probability th…

自旋系统中方差与熵分解的组合方法

Combinatorial Approach for Factorization of Variance and Entropy in Spin Systems

提出基于平衡分隔子递归块分解的组合框架，建立自旋系统方差与熵的近似张量化，并推出 Glauber 动力学在树宽图和有界度平面图上的新快速混合界。

Subexponential mixing for partition chains on grid-like graphs

We consider the problem of generating uniformly random partitions of the vertex set of a graph such that every piece induces a connected subgraph. For the case where we want to have partitions with linearly many pieces of bounded size, we obtain approximate sampling algorithms based on Glauber dynamics which are fixed-parameter tractable with respect to the bandwidth of $G$, with simple-exponential dependence on the bandwidth. For example, for rectangles of constant or logarithmic width this gives polynomial-time sampling algorithms. More generally, this gives sub-exponential algorithms for bo…

Sampling Random Colorings of Sparse Random Graphs

We study the mixing properties of the single-site Markov chain known as the Glauber dynamics for sampling $k$-colorings of a sparse random graph $G(n,d/n)$ for constant $d$. The best known rapid mixing results for general graphs are in terms of the maximum degree $Δ$ of the input graph $G$ and hold when $k>11Δ/6$ for all $G$. Improved results hold when $k>αΔ$ for graphs with girth $\geq 5$ and $Δ$ sufficiently large where $α\approx 1.7632\ldots$ is the root of $α=\exp(1/α)$; further improvements on the constant $α$ hold with stronger girth and maximum degree assumptions. For sparse random grap…

分数对数凹与扇区稳定多项式：计数平面匹配及更多问题

Fractionally Log-Concave and Sector-Stable Polynomials: Counting Planar Matchings and More

针对平面图非完美匹配计数问题，提出了基于多位点Glauber动力学的FPRAS，并引入扇区稳定性和分数对数凹性概念分析其混合性质，为谱独立性提供了基于多项式几何的新工具。