Mixing rate exponent of planar Fortuin-Kasteleyn percolation

Duminil-Copin and Manolescu (2022) recently proved the scaling relations for planar Fortuin-Kasteleyn (FK) percolation. In particular, they showed that the one-arm exponent and the mixing rate exponent are sufficient to derive the other near-critical exponents. The scaling limit of critical FK percolation is conjectured to be a conformally invariant random collection of loops called the conformal loop ensemble (CLE). In this paper, we define the CLE analog of the mixing rate exponent. Assuming the convergence of FK percolation to CLE, we show that the mixing rate exponent for FK percolation agrees with that of CLE. We prove that the CLE$_\kappa$ mixing rate exponent equals $\frac{3 \kappa}{8}-1$, thereby answering Question 3 of Duminil-Copin and Manolescu (2022). The derivation of the CLE exponent is based on an exact formula for the Radon-Nikodym derivative between the marginal laws of the odd-level and even-level CLE loops, which is obtained from the coupling between Liouville quantum gravity and CLE.