Boundary four-point connectivities of conformal loop ensembles

Abstract

We derive the boundary four-point Green's functions for conformal loop ensembles (CLE) with $\hat{I}º\in (4,8)$. Specializing to $\hat{I}º=6$ and $\hat{I}º=16/3$, we establish the exact formulas for the boundary four-point connectivities in critical Bernoulli percolation and the FK-Ising model conjectured by Gori-Viti (2017, 2018). In particular, we identify a logarithmic singularity in the critical FK-Ising model. Our approach also applies to the one-bulk and two-boundary connectivities of CLE, thereby extending the factorization formula of Beliaev-Izyurov (2012) to all $\hat{I}º\in (4,8)$.

38 pages, 1 figure