Gefei Cai

Gefei Cai

PhD Candidate, PKU

Publications

Journal Articles

SLE Loop Measure and Liouville Quantum Gravity

arXiv e-print arXiv:2409.16547, 2024

As recently shown by Holden and two of the authors, the conformal welding of two Liouville quantum gravity (LQG) disks produces a canonical variant of SLE curve whose law is called the SLE loop measure. In this paper, we demonstrate how LQG can be used to study the SLE loop measure. Firstly, we show that for κ∈(8/3,8), the loop intensity measure of the conformal loop ensemble agrees with the SLE loop measure as defined by Zhan (2021). The former was initially considered by Kemppainen and Werner (2016) for κ∈(8/3,4], and the latter was constructed for κ∈(0,8). Secondly, we establish a duality for the SLE loop measure between κ and 16/κ. Thirdly, we obtain the exact formula for the moment of the electrical thickness for the shape (probability) measure of the SLE loop, which in the regime κ∈(8/3,8) was conjectured by Kenyon and Wilson (2004). This relies on the exact formulae for the reflection coefficient and the one-point disk correlation function in Liouville conformal field theory. Finally, we compute several multiplicative constants associated with the SLE loop measure, which are not only of intrinsic interest but also used in our companion paper relating the conformal loop ensemble to the imaginary DOZZ formulae.

Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond

arXiv e-print arXiv:2107.01788v3, 2024

The scaling limit of the probability that n points are on the same cluster for 2D critical percolation is believed to be governed by a conformal field theory (CFT). Although this is not fully understood, Delfino and Viti (2010) made a remarkable prediction on the exact value of a properly normalized three-point probability. It is expressed in terms of the imaginary DOZZ formula of Schomerus, Zamolodchikov and Kostov-Petkova, which extends the structure constants of minimal model CFTs to continuous parameters. Later, similar conjectures were made for scaling limits of random cluster models and O(n) loop models, representing certain three-point observables in terms of the imaginary DOZZ formula. Since the scaling limits of these models can be described by the conformal loop ensemble (CLE), such conjectures can be formulated as exact statements on CLE observables. In this paper, we prove Delfino and Viti’s conjecture on percolation as well as a conjecture of Ikhlef, Jacobsen and Saleur (2015) on the nesting loop statistics of CLE. Our proof is based on the coupling between CLE and Liouville quantum gravity on the sphere, and is inspired by the fact that after reparametrization, the imaginary DOZZ formula is the reciprocal of the three-point function of Liouville CFT. Recently, Nivesvivat, Jacobsen and Ribault systematically studied a CFT with a large class of CLE observables as its correlation functions, including the ones from these two conjectures. We believe that our framework admits sufficient flexibility to exactly solve the three-point functions for CLE observables with natural geometric interpretations, including those from this CFT. As a demonstration, we solve the case corresponding to three points lying on the same loop, where the answer is a variant of the imaginary DOZZ formula.

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

Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, in press, 2024

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.