带孔威尔逊 - 演化集与大规模基尔霍夫森林的根恒等式

中文摘要 AI 辅助

我们为有限可逆马尔可夫链构建了一种带孔威尔逊 - 演化集耦合。变换后的演化集在从带孔域退出之前携带马尔可夫轨迹，而普通演化集在去偏后给出相应概率。应用于带指数杀伤的威尔逊算法时，终点投影产生根定律以及几个指定顶点为根的概率的有序因式分解。我们将此因式分解与已知行列式根公式的概率舒尔补分解相识别。生存投影产生杀伤前击中概率的演化集表示。在满足高斯热核上界的多项式增长图上，在大于四维的维度中，我们得到两点森林连通性在尺度$q^{-1/2}$处的指数局域化，直至自然有限体积修正，且有界$\mathbb E[|C_q(x)|]\leq Cq^{-2}$。相继带孔域中的狄利克雷特征值也为指定根事件给出乘积界。我们记录了根数量的无行列式泊松型集中界，同时明确行列式描述给出更精确的伯努利分解。对完全图进行了精确计算，并将离散环面、瓶颈图和超立方体作为示例处理。

英文摘要

We construct a punctured Wilson--evolving-set coupling for finite reversible Markov chains. The transformed evolving set carries a Markov trajectory up to its exit from a punctured domain, whereas the ordinary evolving set gives the corresponding probabilities after de-biasing. Applied to Wilson's algorithm with exponential killing, the final-point projection yields the root law and an ordered factorization of the probability that several prescribed vertices are roots. We identify this factorization with a probabilistic Schur-complement decomposition of the known determinantal root formula. The survival projection yields an evolving-set representation of hitting probabilities before killing. This representation gives a quantitative consequence which does not follow from the root process alone. On graphs of polynomial growth satisfying a Gaussian heat-kernel upper bound, in dimension larger than four, we obtain exponential localization at scale $q^{-1/2}$ for two-point forest connectivity, up to the natural finite-volume correction, and the bound $\mathbb E[|C_q(x)|]\leq Cq^{-2}$. Dirichlet eigenvalues in successively punctured domains also give product bounds for prescribed root events. We record a determinant-free Poisson-type concentration bound for the number of roots, while making explicit that the determinantal description gives the sharper Bernoulli decomposition. The complete graph is computed exactly and discrete tori, bottleneck graphs and the hypercube are treated as examples.

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30 pages, no figures