Simple Conformal Loop Ensembles on Liouville Quantum Gravity

Abstract

We show that when one draws a simple conformal loop ensemble (CLEκ for κ ∈ (8/3, 4)) on an independent √ κ-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corresponding to the interior of the CLE loops) that are cut out form a Poisson point process of quantum disks. This construction allows us to make direct links between CLE on LQG, asymmetric (4/κ)-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be − cos(4π/κ), which can be interpreted as a "density" of CLE loops in the CLE on LQG setting. Positive jumps correspond to the discovery of a CLE loop (where the LQG length of the loop is given by the jump size) and negative jumps correspond to the moments where the discovery process splits the remaining to be discovered domain into two pieces. Some consequences of this result are the following: (i) It provides a construction of a CLE on LQG as a patchwork/welding of quantum disks. (ii) It allows to construct the "natural quantum measure" that lives in a CLE carpet. (iii) It enables us to derive some new properties and formulas for SLE processes and CLE themselves (without LQG) such as the exact distribution of the trunk of the general asymmetric SLEκ(κ − 6) processes.The present work deals directly with structures in the continuum and makes no reference to discrete models, but our calculations match those for scaling limits of O(N ) models on planar maps with large faces and CLE on LQG. Indeed, our Lévy-tree descriptions are exactly the ones that appear in the study of the large-scale limit of peeling of discrete decorated planar maps such as in recent work of Bertoin, Budd, Curien and Kortchemski.The case of non-simple CLEs on LQG will be the topic of another paper. Contents 1. Introduction 1 2. Background 11 3. Exploring CLEs on quantum half-planes 17 4. CLE explorations of quantum disks 23 5. The natural quantum measure in the CLE carpet 27 Appendix A. Proof of Lemma 3.3 34 References 36

Self-similar growth-fragmentation processes with types

“…CLE explorations of a quantum disc. This example is based on the work of Miller, Sheffield and Werner [MSW20]. We review their results and reformulate them in the multitype setting.…”

“…The paper [MSW20] constructs through these CPI branches a growth-fragmentation process, which is the continuum analogue of the one arising from peeling explorations of Boltzmann planar maps [BBCK18]. Consider some CPI branch towards some point z in the CLE κ carpet, and parametrise it by its quantum length.…”

“…One should then obtain a multitype growth-fragmentation process X γ . However, a consequence of [MSW20, Theorem 1.1] is that these surfaces have the same law (see also [MSW20,Theorem 1.4] for an analogous statement in the quantum half-plane case). This means that, although X γ contains more information than X γ , the features of X γ can be deduced from those of X γ .…”

“…This means that, although X γ contains more information than X γ , the features of X γ can be deduced from those of X γ . Such is for example the case of the martingales involved in Section 3.2, which are the same as those appearing in [MSW20].…”

Tightness of approximations to the chemical distance metric for simple conformal loop ensembles

“…We will thus want to consider a type of exploration which avoids revealing the exact shape of the explored region as much as possible. For this reason, we will explore the CLE κ using CPIs in a "quantum" manner using the relationship between SLE and Liouville quantum gravity (LQG) developed in [41,9] as well as the relationship between CLE and LQG developed in [33]. The reason for this is that exploring a CLE in a quantum manner is analogous to exploring a random planar map using a peeling process and in such an exploration the regions which are cut out depend only on the exploration through their boundary length.…”