2026/07/29 by Zhengwen Qiao
Mathematics · #math.PR
This paper contains critical errors in key derivations. The authors have decided to withdraw it for substantial revision and resubmission
arxiv created 2026/07/31 · arxiv updated 2026/08/03
Let D be a bounded analytic Jordan domain and let γ be chordal SLE2 in D, equipped with its 5/4-dimensional natural-content measure μγ. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let \mathcal Mεγ be the root-intensity measure of loops with duration in [ε2,t0] whose traces hit γ. For every f∈ Cc(D), we prove that ε5/4\mathcal Mεγ(f) converges in L1 to (4)/(5π) vBBμγ(f), where vBB∈(0,∞) is the mean specific area swept out by the Brownian-bridge offset of a natural-time two-sided whole-plane SLE2. 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.