The bulk one-arm exponent for the CLE$_{\kappa'}$ percolations

The conformal loop ensemble (CLE) is a conformally invariant random collection of loops.

In the non-simple regime $\kappa'\in (4,8)$, it describes the scaling limit of the critical Fortuin-Kasteleyn (FK) percolations.

CLE percolations were introduced by Miller-Sheffield-Werner (2017).

The CLE$_{\kappa'}$ percolations describe the scaling limit of a natural variant of the FK percolation called the fuzzy Potts model, which has an additional percolation parameter $r$.

Based on CLE percolations and assuming that the convergence of the FK percolation to CLE, K{ö}hler-Schindler and Lehmkuehler (2022) derived all the arm exponents for the fuzzy Potts model except the bulk one-arm exponent.

In this paper, we exactly solve this exponent, which prescribes the dimension of the clusters in CLE$_{\kappa'}$ percolations.

As a special case, the bichromatic one-arm exponent for the critical 3-state Potts model should be $4/135$.

To the best of our knowledge, this natural exponent was not predicted in physics.

Our derivation relies on the iterative construction of CLE percolations from the boundary conformal loop ensemble (BCLE), and the coupling between Liouville quantum gravity and SLE curves.

The source of the exact solvability comes from the structure constants of boundary Liouville conformal field theory.

A key technical step is to prove a conformal welding result for the target-invariant radial SLE curves.

As intermediate steps in our derivation, we obtain several exact results for BCLE in both the simple and non-simple regimes, which extend results of Ang-Sun-Yu-Zhuang (2024) on the touching probability of non-simple CLE.

This also provides an alternative derivation of the relation between the BCLE parameter $\rho$ and the additional percolation parameter $r$ in CLE percolations, which was originally due to Miller-Sheffield-Werner (2021, 2022).