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

TL;DR

This paper establishes the uniqueness of generalized conformal restriction measures and Malliavin-Kontsevich-Suhov measures for central charge c in (0,1], using a probabilistic approach linked to Brownian loop soups.

Contribution

It introduces a unified probabilistic method for proving measure uniqueness, extending previous results and providing new insights into loop measures for c in (0,1].

Findings

Proves uniqueness of conformal restriction measures for c in (0,1]

Links MKS measures to Brownian loop soup observables

Offers probabilistic insights beyond previous CFT-based approaches

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.