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

Abstract

Let $D$ be a bounded analytic Jordan domain and let ${\hat{I}}^3$ be chordal ${\mathrm{SLE}}_2$ in $D$, equipped with its $5/4$-dimensional natural-content measure $\hat{I}{¼}_{\hat{I}}^3$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let ${\mathcal{M}}_{\epsilon }^3$ be the root-intensity measure of loops with duration in $[{\epsilon }^2,t_0]$ whose traces hit ${\hat{I}}^3$. For every $f\in C_c(D)$, we prove that ${\epsilon }^{5/4}{\mathcal{M}}_{\epsilon }^3(f)$ converges in $L^1$ to $\frac{4}{5\ddot{I}€}{\bar{v}}_{\mathrm{BB}}\hat{I}{¼}_{\hat{I}}^3(f)$, 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.

This paper contains critical errors in key derivations. The authors have decided to withdraw it for substantial revision and resubmission

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