Uniqueness of generalized conformal restriction measures and Malliavin–Kontsevich–Suhov measures for c∈(0,1]

In this paper, we present a unified approach to establish the uniqueness of generalized conformal restriction measures with central charge c∈(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∈(0,1], which was recently obtained in (Baverez and Jego (2024)) for all c≤1 from a CFT framework of SLE loop measures. In contrast, though only valid for c∈(0,1], our approach provides additional probabilistic insights, as it directly links natural quantities of MKS measures to loop-soup observables.