Exact Hausdorff measure of SLE boundary contacts

Abstract

The dimension of a random fractal specifies its scaling exponent; a finite, nonzero measure requires a finer calibration. We determine this calibration for the boundary contacts of chordal SLEκ, 4 < κ < 8. With δ = 2 − 8/κ and β = 8/κ − 1, the exact Hausdorff gauge is h(r) = rδ[log log(1/r)]β. The associated Hausdorff measure on the entire boundary contact set equals a deterministic positive constant times the canonical boundary measure, simultaneously for all Borel sets. The exponent β is sharp: smaller iterated logarithmic powers give zero measure and larger powers give infinite measure on sets of positive finite canonical mass. The proof combines an explicit product bound for Green functions of every order with estimates uniform over the past at actual contact stopping times. An inverse boundary mass clock yields recurrence along one curve, and a covering argument removes all exceptional contacts at zero Hausdorff cost. The resulting identity recovers the canonical boundary mass, and its completed clock, directly from Euclidean geometry.