Duality of random planar maps via percolation

Curien, Nicolas; Richier, Loïc

doi:10.5802/aif.3369

“…In some sense, our results show that from the perspective of the continuum models that should appear in the scaling limit when one considers O(N ) models on well-chosen planar maps (or related models), the features that allowed physicists to use their quantum gravity ideas are indeed valid. It should therefore not be surprising that some of our formulas mirror results that appear in the study of some special planar maps, such as the ones arising in [10,13,5] (see also [11,14] for further related results on the planar maps side). This can be explained by the fact that some of the discrete peeling type processes used in the study of planar maps should indeed give rise to these loops on trunk processes on independent LQG in the scaling limit.…”

Section: Results Of the Present Papersupporting

confidence: 68%

Simple Conformal Loop Ensembles on Liouville Quantum Gravity

Miller¹,

Sheffield⋆²,

Werner³

2020

Preprint

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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

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“…Many aspects of the arguments will mirror those of our paper for -decorated LQG surfaces [ 39 ] (for ), that we will also directly refer to for an introduction and background. Just as in [ 39 ], all our arguments take place in the continuum and do not build on any considerations about random decorated planar maps, but the results do mirror some of the results that appear when one studies O ( N )-models or FK-percolation models on planar maps via enumerative techniques, such as in [ 4 – 6 , 8 , 10 ]. The Markovian structure that we unveil in the present paper can be viewed as the continuum counterpart on the peeling algorithms and their properties for these discrete models.…”

Section: Introductionmentioning

confidence: 53%

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

Miller

¹

,

Sheffield⋆

²

,

Werner

³

2021

Probab. Theory Relat. Fields

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We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ for $$\kappa '$$ κ ′ in (4, 8) that is drawn on an independent $$\gamma $$ γ -LQG surface for $$\gamma ^2=16/\kappa '$$ γ 2 = 16 / κ ′ . The results are similar in flavor to the ones from our companion paper dealing with $$\hbox {CLE}_{\kappa }$$ CLE κ for $$\kappa $$ κ in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ independently into two colors with respective probabilities p and $$1-p$$ 1 - p . This description was complete up to one missing parameter $$\rho $$ ρ . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and $$\kappa '$$ κ ′ . It shows in particular that $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$ CLE 16 / κ ′ are related via a continuum analog of the Edwards-Sokal coupling between $$\hbox {FK}_q$$ FK q percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if $$q=4\cos ^2(4\pi / \kappa ')$$ q = 4 cos 2 ( 4 π / κ ′ ) . This provides further evidence for the long-standing belief that $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$ CLE 16 / κ ′ represent the scaling limits of $$\hbox {FK}_q$$ FK q percolation and the q-Potts model when q and $$\kappa '$$ κ ′ are related in this way. Another consequence of the formula for $$\rho (p,\kappa ')$$ ρ ( p , κ ′ ) is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

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“…Many aspects of the arguments will mirror those of our paper for CLE κ -decorated LQG surfaces [39] (for κ ∈ (8/3, 4)), that we will also directly refer to for an introduction and background. Just as in [39], all our arguments take place in the continuum and do not build on any considerations about random decorated planar maps, but the results do mirror some of the results that appear when one studies O(N )-models or FK-percolation models on planar maps via enumerative techniques, such as in [5,8,4,6,10]. The Markovian structure that we unveil in the present paper can be viewed as the continuum counterpart on the peeling algorithms and their properties for these discrete models.…”

Section: Introductionmentioning

confidence: 63%

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

Miller¹,

Sheffield⋆²,

Werner³

2020

Preprint

0

View full text Add to dashboard Cite

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE κ for κ in (4, 8) that is drawn on an independent γ-LQG surface for γ 2 = 16/κ . The results are similar in flavor to the ones from our paper [39] dealing with CLEκ for κ in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the CLE κ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps.This has consequences for questions that do a priori not involve LQG surfaces: Our previous paper [37] described the law of interfaces obtained when coloring the loops of a CLE κ independently into two colors with respective probabilities p and 1 − p. This description was complete up to one missing parameter ρ. The results of the present paper about CLE on LQG allow us to determine its value in terms of p and κ .It shows in particular that CLE κ and CLE 16/κ are related via a continuum analog of the Edwards-Sokal coupling between FKq percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if q = 4 cos 2 (πκ /4). This provides further evidence for the long-standing belief that CLE κ and CLE 16/κ represent the scaling limits of FKq percolation and the q-Potts model when q and κ are related in this way.Another consequence of the formula for ρ(p, κ ) is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

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Duality of random planar maps via percolation

Abstract: Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-σmersenne.org

Cited by 14 publications

References 30 publications

Simple Conformal Loop Ensembles on Liouville Quantum Gravity

Simple Conformal Loop Ensembles on Liouville Quantum Gravity

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

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