We prove that the single-site Glauber dynamics for sampling proper $q$-colorings mixes in $O_\Delta(n\log n)$ time on line graphs with $n$ vertices and maximum degree $\Delta$ when $q>(1+o(1))\Delta$. The main tool in our proof is the matrix trickle-down theorem developed by Abdolazimi, Liu and Oveis Gharan (FOCS, 2021).

翻译：我们证明，对于具有$n$个顶点和最大度$\Delta$的线图，当$q>(1+o(1))\Delta$时，用于采样适定$q$着色的单点Glauber动力学在$O_\Delta(n\log n)$时间内混合。证明的主要工具是Abdolazimi、Liu和Oveis Gharan（FOCS, 2021）发展的矩阵滴流定理。