Uniqueness of generalized conformal restriction measures and Malliavin-Kontsevich-Suhov measures for $c \in (0,1]$

TL;DR

The work delivers a unified probabilistic approach to the uniqueness of generalized conformal restriction measures for $c\in(0,1]$ by linking them to Brownian loop soup and $\mathrm{SLE}_{8/3}$ loop soup observables, and extends this framework to the Malliavin-Kontsevich-Suhov loop measures. It provides explicit loop-soup representations for non-intersection probabilities and MKS masses, establishes existence/uniqueness for both chordal and radial restriction measures, and derives a probabilistic formula for the MKS mass via conformal derivatives. The results recover and illuminate SLE reversibility and duality from restriction-measure uniqueness, connect radial restriction to $SLE_\kappa(\rho)$-CLE constructions, and introduce the electrical-thickness concept linking loop-soup geometry to conformal maps. Overall, the paper enhances the probabilistic understanding of restriction measures, complements CFT approaches, and opens avenues for extensions to other geometries and regime parameters.

Abstract

In this paper, we present a unified approach to establish the uniqueness of generalized conformal restriction measures with central charge $c \in (0, 1]$ in both chordal and radial cases, by relating these measures to the Brownian loop soup. Our method also applies to the uniqueness of the Malliavin-Kontsevich-Suhov loop measures for $c \in (0,1]$, which was recently obtained in [Baverez-Jego, arXiv:2407.09080] for all $c \leq 1$ from a CFT framework of SLE loop measures. In contrast, though only valid for $c \in (0,1]$, our approach provides additional probabilistic insights, as it directly links natural quantities of MKS measures to loop-soup observables.

Theorems & Definitions (63)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4: kontsevich2007malliavin
• Theorem 1.5
• Theorem 1.6
• Proposition 1.7: werner2013cle
• proof
• Proposition 1.8
• Theorem 1.9