For general spin systems, we prove that a contractive coupling for any local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics, arbitrary heat-bath block dynamics, and the Swendsen-Wang dynamics. This reveals a novel connection between probabilistic techniques for bounding the convergence to stationarity and analytic tools for analyzing the decay of relative entropy. As a corollary of our general results, we obtain $O(n\log{n})$ mixing time and $\Omega(1/n)$ modified log-Sobolev constant of the Glauber dynamics for sampling random $q$-colorings of an $n$-vertex graph with constant maximum degree $\Delta$ when $q > (11/6 - \epsilon_0)\Delta$ for some fixed $\epsilon_0>0$. We also obtain $O(\log{n})$ mixing time and $\Omega(1)$ modified log-Sobolev constant of the Swendsen-Wang dynamics for the ferromagnetic Ising model on an $n$-vertex graph of constant maximum degree when the parameters of the system lie in the tree uniqueness region. At the heart of our results are new techniques for establishing spectral independence of the spin system and block factorization of the relative entropy. On one hand we prove that a contractive coupling of a local Markov chain implies spectral independence of the Gibbs distribution. On the other hand we show that spectral independence implies factorization of entropy for arbitrary blocks, establishing optimal bounds on the modified log-Sobolev constant of the corresponding block dynamics.
翻译:对于一般旋转系统,我们证明,任何本地马可夫链的契约结合意味着混合时间的最佳界限,以及包括Glauber动态、任意热吸浴区块动态和Swindsen-Wang动态在内的大型马尔科夫链的修改的日志-Sobolev常数,包括Glauber动态、任意热吸浴区块动态和Swindsen-Wang动态。这揭示了将趋同于静态的概率技术与分析相对变异的工具之间的新联系。作为我们总体结果的一个必然结果,我们获得了 $(n\log{n}) 混合时间和$(Omega(1/n) 美元) 的修改的日志-Sobolev值常数常数常数常数常数。 当 $ > (11/6 - \ \ \ \ \ \ epsillon_0)\ Delta$ 用于某些固定的 $\ epsilon_0 > 美元。我们还获得了 commluslusloral liveral livalal listral liver liver liver liver liver liver reslistral resal resal restial resml resml resmal restial restial restial resmlview resmlval res resm resmlation resmlation resmlation resm resmlation resmlation resmlation resmlutututut ressmlation ressml,我们的系统的系统, 当值系统显示一个固定的系统的固定的系统的内基的内基的内基的内基的内值的内值, 当值,我们等的内基的内基的内基的内基的内基的内值的内基值的内值的内值的内值的内值的内基的内基的内值,我们的内基的内基的内基的内值。