块耦合与快速混合k-heights
Block coupling and rapidly mixing k-heights
AI总结：
本文研究了图上k-heights的Markov chain，提出了一种基于单调耦合的block coupling新技术，并证明了该链在2-heights和3-heights情况下于特定网格状图和平面三次3-连通图上快速混合。
摘要：
图$G=(V, E)$上的$k$-heights是一种赋值$V\to\{0, \ldots, k\}$，使得相邻顶点上的值相差不超过$1$。我们研究了$k$-heights上的Markov chain，该链在每一步随机选择一个顶点，并在可行时将该顶点的值增加或减少$1$。在$2$-heights和$3$-heights的情况下，我们证明了该Markov chain在特定类网格状图和平面三次$3$-连通图上快速混合。该结果基于一种名为block coupling的新技术，该技术源自成熟的单调耦合方法。在分析其他作用于形成分配格的自旋系统配置的Markov chain时，该技术也可能有效，因此具有独立的意义。
英文摘要：
A $k$-height on a graph $G=(V, E)$ is an assignment $V\to\{0, \ldots, k\}$ such that the value on ajacent vertices differs by at most $1$. We study the Markov chain on $k$-heights that in each step selects a vertex at random, and, if admissible, increases or decreases the value at this vertex by one. In the cases of $2$-heights and $3$-heights we show that this Markov chain is rapidly mixing on certain families of grid-like graphs and on planar cubic $3$-connected graphs. The result is based on a novel technique called block coupling, which is derived from the well-established monotone coupling approach. This technique may also be effective when analyzing other Markov chains that operate on configurations of spin systems that form a distributive lattice. It is therefore of independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
