Simple conformal loop ensembles on Liouville quantum gravity

We show that, when one draws a simple conformal loop ensemble (CLE? for ??(8/3,4)) on an independent ?-Liouville quantum gravity (LQG) surface and explores the CLE in natural Markovian way, surfaces (e.g., corresponding to interior of loops) that are cut out form Poisson point process disks. This construction allows us make direct links between LQG, (4/?)-stable processes, labeled branching trees. The ratio positive negative jump intensities these processes turns be ?cos(4?/?) which can interpreted as “density” loops LQG setting. Positive jumps correspond discovery (where length is given by size) moments where splits remaining discovered domain into two pieces. Some consequences this result following: (i) It provides patchwork/welding (ii) construct “natural measure” lives carpet. (iii) enables derive some new properties formulas SLE themselves (without LQG) such exact distribution trunk general SLE?(??6) processes. present work deals directly with structures continuum makes no reference discrete models, but our calculations match those scaling limits O(N) models planar maps large faces LQG. Indeed, Lévy-tree descriptions exactly ones appear study large-scale limit peeling decorated maps, recent Bertoin, Budd, Curien Kortchemski. case nonsimple CLEs studied another paper.