AI 中文总结
研究引入布朗环捕手，它在LERW和布朗轨迹间插值，为特定中心电荷的布朗环汤簇提供规范延续，给出其恢复性质、外边界特征等，通过有限图上随机游走环捕手经格点近似收敛得到，还提出格林函数测试并扩展到三维。
AI 中文摘要
我们引入了一族平面布朗运动的随机连通闭子集，称为布朗环捕手，它在连续统环擦除随机游走（LERW）和布朗轨迹之间进行插值。这为中心电荷为\(-2\leq c<0\)的布朗环汤簇提供了规范延续。对于每个这样的\(c\)，相应的环捕手满足以下恢复性质：添加强度为\(-c/2\)的独立布朗环汤中与它相交的所有环可恢复布朗轨迹。此外，\(c<-2\)时不存在这样的定律。我们还表明其外边界局部是SLE\(_\kappa\)，其中\(\kappa=\frac{1}{3}(13 - c - \sqrt{(1 - c)(25 - c)})\in[2,\frac{8}{3})\)，当\(-2 < c < 0\)时，它与半径为\(\varepsilon\)的内球相交的概率渐近地与\(|\log\varepsilon|^{-1+\frac{c}{2}}\)成比例。我们的构造从任何有限图上的随机游走环捕手开始，其定律由有限线性系统确定。我们证明对于\(-2\leq c<0\)其解是非负的，而\(c<-2\)时非负性可能不成立。关键要素是一种新的纠缠多路径LERW，当用单个公共随机游走环汤装饰时可恢复独立随机游走路径的并集。然后我们证明在格点近似下随机游走环捕手收敛到布朗环捕手。为此，我们提出了一种新颖的格林函数测试，基于纠缠多路径LERW将布朗环捕手的恢复性质转化为剩余域中格林函数的所有混合矩。因此，恢复性质不仅刻画了布朗环捕手的填充，还刻画了其完整定律。格林函数测试也扩展到了三维情况。
英文摘要
We introduce a family of random connected closed subsets of planar Brownian motion, called Brownian loop-catchers, which interpolate between the continuum loop-erased random-walk (LERW) and the Brownian trace. This provides the canonical continuation of Brownian loop soup clusters to central charges $-2\le c<0$. For each such $c$, the corresponding loop-catcher satisfies the following recovery property: adding all loops from an independent Brownian loop soup of intensity $-c/2$ that intersect it recovers the Brownian trace. Furthermore, no such law exists for $c<-2$. We also show that its outer boundary is locally SLE$_κ$ with $κ= \frac{1}{3}\left(13 - c - \sqrt{(1-c)(25-c)}\right)\in[2,\frac83)$, and the probability that it intersects an interior ball of radius $\varepsilon$ is asymptotically proportional to $|\log\varepsilon|^{-1+\frac{c}{2}}$ when $-2<c<0$. Therefore, a planar Brownian trace contains an SLE$_κ$-type curve for every $κ\in[2,\frac83]$. Our construction begins with a random-walk loop-catcher on any finite graph such that recursively inserting loops from an independent random-walk loop soup to it recovers the original random walk. The key ingredient is a new entangled multipath LERW, which recovers a union of independent random-walk paths when decorated with a single common random-walk loop soup. We then prove that the random-walk loop-catcher converges to the Brownian loop-catcher under lattice approximations. To this end, we propose a novel Green function test which converts the recovery property of the Brownian loop-catcher into all mixed moments of Green functions in the remaining domain, based on the entangled multipath LERW. Consequently, the recovery property characterizes the full law of the Brownian loop-catcher, not only its filling. The Green function test also extends to the three-dimensional case.
Comments45 pages, 2 figures. Added time parametrization for the random-walk loop-catcher
