亚临界平面布朗环汤中的多窗口轨迹连通性
Multi-window trace connectivity in subcritical planar Brownian loop soups
AI总结：
研究亚临界平面布朗环汤中相交环轨迹簇与固定数量分离收缩体圆盘相交概率，通过多目标布朗环测度估计等方法，得出该概率在对数指数有损失时是单臂概率乘积的结论，部分成果源于前人。
AI中文摘要：
设\(\cL_D^\theta\)为有界光滑平面区域中强度\(0<\theta<1/2\)的布朗环汤。我们估计相交环轨迹的一个簇与固定数量的分离收缩体圆盘相交的概率。对于每个固定的\(q\geq3\)，此概率在对数指数上有任意损失的情况下，是相应单臂概率的乘积。上界不是通过成对估计相乘或形式上应用BK得到的。我们对不局限于一个目标领圈的环进行条件设定，在每个目标处将局部臂充电到其穿透半径，并通过多目标布朗环测度估计来控制联合穿透深度。后者源于有限被杀网络上的标记舒尔补展开、均匀平面容量估计和随机游走 - 环汤耦合。下界使用缠绕分隔符、一个宏观桥环和泊松FKG。单窗口指数和双窗口估计归功于杰戈、卢普和钱。
英文摘要：
Let \(\cL_D^θ\) be a Brownian loop soup of intensity \(0<θ<1/2\) in a bounded smooth planar domain. We estimate the probability that one cluster of intersecting loop traces meets a fixed number of separated shrinking bulk discs. For every fixed \(q\ge3\), this probability is, up to an arbitrary loss in the logarithmic exponents, the product of the corresponding one-arm probabilities. The upper bound is not obtained by multiplying pairwise estimates or by a formal application of BK. We condition on the loops not confined to one target collar, charge a local arm down to their penetration radius at every target, and control the joint penetration depths by a multi-target Brownian loop-measure estimate. The latter follows from a marked Schur-complement expansion on finite killed networks, uniform planar capacity estimates, and a random-walk-loop-soup coupling. The lower bound uses winding separators, one macroscopic bridge loop, and Poisson FKG. The one-window exponent and the two-window estimate are due to Jego, Lupu and Qian.
