A common obstruction to efficient sampling from high-dimensional distributions is the multimodality of the target distribution because Markov chains may get trapped far from stationarity. Still, one hopes that this is only a barrier to the mixing of Markov chains from worst-case initializations and can be overcome by choosing high-entropy initializations, e.g., a product or weakly correlated distribution. Ideally, from such initializations, the dynamics would escape from the saddle points separating modes quickly and spread its mass between the dominant modes. In this paper, we study convergence from high-entropy initializations for the random-cluster and Potts models on the complete graph -- two extensively studied high-dimensional landscapes that pose many complexities like discontinuous phase transitions and asymmetric metastable modes. We study the Chayes--Machta and Swendsen--Wang dynamics for the mean-field random-cluster model and the Glauber dynamics for the Potts model. We sharply characterize the set of product measure initializations from which these Markov chains mix rapidly, even though their mixing times from worst-case initializations are exponentially slow. Our proofs require careful approximations of projections of high-dimensional Markov chains (which are not themselves Markovian) by tractable 1-dimensional random processes, followed by analysis of the latter's escape from saddle points separating stable modes.

翻译：高维分布高效采样的常见障碍是目标分布的多模态性，因为马尔可夫链可能被困在远离平稳分布的地方。然而，人们希望这仅是马尔可夫链在最坏初态下混合的障碍，可以通过选择高熵初态（如乘积分布或弱相关分布）来克服。理想情况下，从这类初态出发，动力学能快速逃离分隔模式的鞍点，并在主导模式间传播质量。本文研究完全图上随机团簇模型与Potts模型从高熵初态出发的收敛性——这两个被广泛研究的高维景观具有诸多复杂性，如非连续相变和非对称亚稳态模式。我们研究了平均场随机团簇模型的Chayes-Machta动力学和Swendsen-Wang动力学，以及Potts模型的Glauber动力学。我们精确刻画了使这些马尔可夫链快速混合的乘积测度初态集合，尽管在最坏初态下其混合时间呈指数级缓慢。我们的证明需要将高维马尔可夫链（其自身非马尔可夫）的投影通过可处理的单维随机过程进行精确近似，随后分析后者逃离分隔稳定模式鞍点的过程。