论文标题
布朗循环群簇的边界上的多个点
Multiple points on the boundaries of Brownian loop-soup clusters
论文作者
论文摘要
对于单位磁盘中的强度$ c \ in（0,1] $中的布朗环汤汤，我们可以肯定的是，几乎可以肯定的是，在其任何簇的任何边界的任何部分上，简单的（分别为double）点都具有hausdorff dimension $ 2-ξ_c（2）$（2）$（分别为$ 2-ξ_c（4）$ cumpution $ 2-ectecountion $ 2-ectecounts $ cuncutiate $ cumport insute in Contuctive insute discounts $ cutement（k）$ ectount（k） ARXIV：1901.05436。 作为中间结果，我们建立了布朗循环汤的分离引理，这是在环境汤的情况下获得对非交流和非混乱概率的尖锐估计的强大工具。特别是，它允许我们定义一个广义交叉指数的家族$ξ_c（k，λ）$，并表明$ξ_c（k）$是限制为$λ\ searrow 0 $ of $ξ_c（k，λ）$。
For a Brownian loop soup with intensity $c\in(0,1]$ in the unit disk, we show that almost surely, the set of simple (resp. double) points on any portion of boundary of any of its clusters has Hausdorff dimension $2-ξ_c(2)$ (resp. $2-ξ_c(4)$), where $ξ_c(k)$ is the generalized disconnection exponent computed in arxiv:1901.05436. As a consequence, when the dimension is positive, such points are a.s. dense on every boundary of every cluster. There are a.s. no triple points on the cluster boundaries. As an intermediate result, we establish a separation lemma for Brownian loop soups, which is a powerful tool for obtaining sharp estimates on non-intersection and non-disconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents $ξ_c(k, λ)$, and show that $ξ_c(k)$ is the limit as $λ\searrow 0$ of $ξ_c(k, λ)$.
