Liouville quantum gravity surfaces with boundary as matings of trees

Ang, Morris; Gwynne, Ewain

doi:10.1214/20-aihp1068

Cited by 16 publications

(36 citation statements)

References 82 publications

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“…Therefore, under QD 1,0 (a) ⊗ CLE κ , conditioning on h (η) = b, the conditional law of the quantum area of A η is given by i≥1 The rest of this section is devoted to the proof of Proposition 4.2. Recall that the law of A i 's are inverse gamma distributions [AG21,ARS21]. Therefore Proposition 4.2 is a statement about stable Lévy processes and inverse gamma distributions.…”

Section: Area Distribution Of the Quantum Annulusmentioning

confidence: 99%

Integrability of the conformal loop ensemble

Ang¹,

Sun²

2021

Preprint

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We demonstrate that the conformal loop ensemble (CLE) has a rich integrable structure by establishing exact formulas for two CLE observables. The first describes the joint moments of the conformal radii of loops surrounding three points for CLE on the sphere. Up to normalization, our formula agrees with the imaginary DOZZ formula due to Zamolodchikov ( 2005) and Kostov-Petkova ( 2007), which is the three-point structure constant of certain conformal field theories that generalize the minimal models. This verifies the CLE interpretation of the imaginary DOZZ formula by Ikhlef, Jacobsen and Saleur (2015). Our second result is for the moments of the electrical thickness of CLE loops first considered by Kenyon and Wilson (2004). Our proofs rely on the conformal welding of random surfaces and two sources of integrability concerning CLE and Liouville quantum gravity (LQG). First, LQG surfaces decorated with CLE inherit a rich integrable structure from random planar maps decorated with the O(n) loop model. Second, as the field theory describing LQG, Liouville conformal field theory is integrable. In particular, the DOZZ formula and the FZZ formula for its structure constants are crucial inputs to our results.

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Section: Area Distribution Of the Quantum Annulusmentioning

confidence: 99%

Integrability of the conformal loop ensemble

Ang¹,

Sun²

2021

Preprint

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“…In [233] the author showed that when two infinite half-plane-homeomorphic 𝛾-LQG surfaces are "conformally welded" to one another along their boundaries -and the new surface is conformally mapped to the half-plane -the interface between these curves becomes an SLE 𝜅 curve with 𝜅 = 𝛾 2 . The sphere version is established in a follow-up work [192], and a disk version appears in [13]. Establishing the correlation formula for 𝜅 > 8 was achieved in With Duplantier and Miller, the author showed the equivalence of the peanosphere approach and the SLE-decorated-LQG approach in [102], which draws from the imaginary geometry results in [188][189][190][191] and the quantum zipper construction in [233].…”

Section: Smirnovmentioning

confidence: 99%

What is a random surface?

Sheffield⋆¹

2022

Preprint

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Given 2𝑛 unit equilateral triangles, there are finitely many ways to glue each edge to a partner. We obtain a random sphere-homeomorphic surface by sampling uniformly from the gluings that produce a topological sphere. As 𝑛 → ∞ these random surfaces (appropriately scaled) converge in law. The limit is a "canonical" sphere-homeomorphic random surface, much the way Brownian motion is a canonical random path.Depending on how the surface space and convergence topology are specified, the limit is the Brownian sphere, the peanosphere, the pure Liouville quantum gravity sphere, or a certain conformal field theory. All of these objects have concise definitions, and are all in some sense equivalent, but the equivalence is highly non-trivial, building on hundreds of math and physics papers over the past half century.More generally, the "continuum random surface embedded in 𝑑-dimensional Euclidean space" makes a kind of sense for 𝑑 ∈ (−∞, 25) even when 𝑑 is not a positive integer; and this can be extended to higher genus surfaces, surfaces with boundary, and surfaces with marked points or other decoration.These constructions have deep roots in both mathematics and physics, drawing from classical graph theory, complex analysis, probability and representation theory, as well as string theory, planar statistical physics, random matrix theory and a simple model for twodimensional quantum gravity.We present here an informal, colloquium-level overview of the subject, which we hope will be accessible to both newcomers and experts. We aim to answer, as cleanly as possible, the fundamental question. What is a random surface?

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“…Using the four algebraic equations found in Step 2, one can also express Z 1 (u, t, ν) as a rational function of Z 0 (u, t, ν), u, t, ν and z 1,0 (t, ν), z 3,0 (t, ν). Then, plug this relation into one of the two loop equations, and we obtain the second equation of (6).…”

Section: Rational Parametrizations Of the Generating Functionsmentioning

confidence: 99%

Ising Model on Random Triangulations of the Disk: Phase Transition

Chen

Turunen

2022

Commun. Math. Phys.

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In Chen and Turunen (Commun Math Phys 374(3):1577–1643, 2020), we have studied the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the infinite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform infinite half-planar triangulation decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of finite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in Chen and Turunen (2020) to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a. Liouville quantum gravity).

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Liouville quantum gravity surfaces with boundary as matings of trees

Cited by 16 publications

References 82 publications

Integrability of the conformal loop ensemble

Integrability of the conformal loop ensemble

What is a random surface?

Ising Model on Random Triangulations of the Disk: Phase Transition

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