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

Abstract

Let $D$ be a bounded analytic Jordan domain and let $\gamma$ be chordal $\mathrm{SLE}_2$ in $D$, equipped with its $5/4$-dimensional natural-content measure $\mu_\gamma$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let $\mathcal M_\varepsilon^\gamma$ be the root-intensity measure of loops with duration in $[\varepsilon^2,t_0]$ whose traces hit $\gamma$. For every $f\in C_c(D)$, we prove that $\varepsilon^{5/4}\mathcal M_\varepsilon^\gamma(f)$ converges in $L^1$ to $\frac{4}{5\pi}\bar v_{\mathrm{BB}}\mu_\gamma(f)$, 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.