Zhengwen Qiao | Searcharxiv

arXiv2026

Multi-window trace connectivity in subcritical planar Brownian loop soups

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.

math.PR↗

arXiv2026

Small Brownian Loops Hitting SLE$_2$: An Exact Natural-Content Limit

Let $D$ be a bounded analytic Jordan domain and let $γ$ be chordal $\mathrm{SLE}_2$ in $D$, equipped with its $5/4$-dimensional natural-content measure $μ_γ$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let $\mathcal M_\varepsilon^γ$ be the root-intensity measure of loops with duration in $[\varepsilon^2,t_0]$ whose traces hit $γ$. For every $f\in C_c(D)$, we prove that $\varepsilon^{5/4}\mathcal M_\varepsilon^γ(f)$ converges in $L^1$ to $\frac{4}{5π}\bar v_{\mathrm{BB}}μ_γ(f)$, where $\bar v_{\mathrm{BB}}\in(0,\infty)$ is the mean specific area swept out by the Brownian-bridge offset of a natural-time two-sided whole-plane $\mathrm{SLE}_2$. In particular, the positive random measures converge vaguely in probability. The proof uses a duration-octave identity, a deterministic finite-$R$ reference coefficient obtained from a stopped two-arm Markov skeleton, a marked physical Palm tangent, an annular remote-return estimate, and a legal mesoscopic diagonal. As an application, the uniformly time-marked roots of an independent Brownian loop soup satisfy the corresponding vague law of large numbers. The analytic-boundary hypothesis enters only through a global finite-domain uniform-integrability estimate; its analogue for an arbitrary bounded Jordan domain remains open.

math.PR↗