Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles

Miller, Jason; Schoug, Lukas

doi:10.48550/arxiv.2201.01748

Cited by 2 publications

(6 citation statements)

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“…Their axioms are slightly different from our Definition 1.9, where they use the Markovian property to replace the locality. Therefore, though the constructions of natural measures in [25] and this paper are similar, the approaches to show the uniqueness are rather different: in [25] they use a martingale argument, while we make heavy use of LQG. It is also worth mentioning that although locality is a priori stronger than the Markovian property as an assumption, it is indeed satisfied by all natural measures (or candidates thereof) for these fractals defined via different approaches.…”

Section: Discussionmentioning

confidence: 99%

“…A key identity that connects the LQG and Euclidean world is the KPZ relation; see Remark 4.7 or Appendix B in [12] for further reference. We alnote that a recent paper [25] also discusses the axiomatic construction on CLE carpet and gasket. Their axioms are slightly different from our Definition 1.9, where they use the Markovian property to replace the locality.…”

Section: Discussionmentioning

confidence: 99%

“…We also remark that our proof can not be easily extended to the critical case κ = 4 for the CLE 4 carpet as the corresponding parameter γ(= √ κ = 2) of LQG is critical. In [25], an approximation approach is offered for the CLE 4 carpet measure.…”

Section: Discussionmentioning

confidence: 99%

“…The existence of Minkowski content for cut points of general SLE κ 's for κ ∈ (4,8) remains open except in a special case, where the work [16] shows the existence of Minkowski content of Brownian cut points as well as for cut points of SLE 6 . Recently, [25] discusses the axiomatic construction on the natural measures of CLE carpet and gasket.…”

Section: Background and Motivationsmentioning

confidence: 99%

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On Natural Measures of SLE- and CLE-Related Random Fractals

Cai¹

2022

Preprint

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In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal points and the CLE carpet/gasket. As an application, we also show the equivalence between our natural measures defined in this paper (i.e. CLE pivotal and gasket measures) and their discrete analogs of counting measures in critical continuum planar Bernoulli percolation in [Garban-Pete-Schramm, J. Amer. Math. Soc., 2013]. Although the existence and uniqueness for the natural measure for CLE carpet/gasket have already been proved in [Miller-Schoug, arXiv:2201.01748], in this paper we provide with a different argument via the coupling of CLE and LQG.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Background and Motivationsmentioning

confidence: 99%

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On Natural Measures of SLE- and CLE-Related Random Fractals

Cai¹

2022

Preprint

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“…LQG coupled with CLE [MSW20, MSW21] are proved or conjectured to describe the scaling limit of random planar maps decorated with these models. This coupling has been used in [AS21,Mil21,MS22] to derive results for CLE itself. As we recall in Section 4, the quantum annulus QA γ is defined by a symmetric modification of the law of the annulus on a quantum disk bounded by an outermost loop and the disk boundary.…”

Section: 3mentioning

confidence: 99%

The moduli of annuli in random conformal geometry

Ang¹,

Rémy²,

Sun³

2022

Preprint

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We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the modulus of the Brownian annulus describing the scaling limit of uniformly sampled planar maps with annular topology, which is as predicted from the ghost partition function in bosonic string theory. The second is for the law of the modulus of the annulus bounded by a loop of a simple conformal loop ensemble (CLE) on a disk and the disk boundary. The formula is as conjectured from the partition function of the O(n) loop model on the annulus derived by Cardy (2006). The third is for the annulus partition function of the SLE 8/3 loop introduced by Werner (2008). It again confirms a prediction of Cardy (2006). The physics principle underlying our proofs is that 2D quantum gravity coupled with conformal matters can be decomposed into three conformal field theories (CFT): the matter CFT, the Liouville CFT, and the ghost CFT. At the technical level, we rely on two types of integrability in Liouville quantum gravity, one from the scaling limit of random planar maps, the other from the Liouville CFT. We expect our method to be applicable to a variety of questions related to the random moduli of non-simply-connected random surfaces.

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Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles

Cited by 2 publications

References 0 publications

On Natural Measures of SLE- and CLE-Related Random Fractals

On Natural Measures of SLE- and CLE-Related Random Fractals

The moduli of annuli in random conformal geometry

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