We address the convergence rate of Markov chains for randomly generating an edge coloring of a given tree. Our focus is on the Glauber dynamics which updates the color at a randomly chosen edge in each step. For a tree $T$ with $n$ vertices and maximum degree $\Delta$, when the number of colors $q$ satisfies $q\geq\Delta+2$ then we prove that the Glauber dynamics has an optimal relaxation time of $O(n)$, where the relaxation time is the inverse of the spectral gap. This is optimal in the range of $q$ in terms of $\Delta$ as Dyer, Goldberg, and Jerrum (2006) showed that the relaxation time is $\Omega(n^3)$ when $q=\Delta+1$. For the case $q=\Delta+1$, we show that an alternative Markov chain which updates a pair of neighboring edges has relaxation time $O(n)$. Moreover, for the $\Delta$-regular complete tree we prove $O(n\log^2{n})$ mixing time bounds for the respective Markov chain. Our proofs establish approximate tensorization of variance via a novel inductive approach, where the base case is a tree of height $\ell=O(\Delta^2\log^2{\Delta})$, which we analyze using a canonical paths argument.

翻译：本文研究了用于随机生成给定树边着色的马尔可夫链的收敛速率。我们重点关注在每一步中随机选择一条边更新其颜色的Glauber动力学。对于具有n个顶点且最大度为Δ的树T，当颜色数q满足q≥Δ+2时，我们证明Glauber动力学具有O(n)的最优弛豫时间，其中弛豫时间是谱隙的倒数。这在q关于Δ的取值范围中是最优的，因为Dyer、Goldberg和Jerrum（2006）已证明当q=Δ+1时弛豫时间为Ω(n³)。针对q=Δ+1的情况，我们证明一种更新相邻边对的替代马尔可夫链具有O(n)的弛豫时间。此外，对于Δ正则完全树，我们证明了相应马尔可夫链具有O(nlog²n)的混合时间边界。我们的证明通过一种新颖的归纳方法建立了方差的近似张量化，其中基础情形是高度为ℓ=O(Δ²log²Δ)的树，我们使用规范路径论证法对其进行分析。