Unit boundary length quantum disk: a study of two different perspectives and their equivalence

@article{Cercle2019UnitBL,
  title={Unit boundary length quantum disk: a study of two different perspectives and their equivalence},
  author={Baptiste Cercl'e},
  journal={arXiv: Probability},
  year={2019}
}
The theory of the 2-dimensional Liouville Quantum Gravity, first introduced by Polyakov in his 1981 work has become a key notion in the study of random surfaces. In a series of articles, David, Huang, Kupiainen, Rhodes and Vargas, on the one hand, and Duplantier, Miller and Sheffield on the other hand, investigated this topic in the realm of probability theory, and both provided definitions for fundamentals objects of the theory: the unit area quantum sphere and the unit boundary length quantum… 
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References

SHOWING 1-10 OF 48 REFERENCES
Two Perspectives of the 2D Unit Area Quantum Sphere and Their Equivalence
Abstract2D Liouville quantum gravity (LQG) is used as a toy model for 4D quantum gravity and is the theory of world-sheet in string theory. Recently there has been growing interest in studying LQG in
Liouville quantum gravity on complex tori
In this paper, we construct Liouville Quantum Field Theory (LQFT) on the toroidal topology in the spirit of the 1981 seminal work by Polyakov [Phys. Lett. B 103, 207 (1981)]. Our approach follows the
Liouville quantum gravity on the unit disk
Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann
Liouville quantum gravity and KPZ
AbstractConsider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫D∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on
Liouville quantum gravity as a mating of trees
There is a simple way to "glue together" a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which
Liouville quantum gravity surfaces with boundary as matings of trees
  • M. Ang, Ewain Gwynne
  • Physics, Mathematics
    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
  • 2021
For $\gamma \in (0,2)$, the quantum disk and $\gamma$-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits
Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $\gamma$, and it has long been
Liouville quantum gravity and the Brownian map I : The QLE ( 8 / 3 , 0 ) metric
Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter γ, and it has long been believed
Liouville quantum gravity and the Brownian map III: the conformal structure is determined
Previous works in this series have shown that an instance of a $$\sqrt{8/3}$$ 8 / 3 -Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure
Gaussian multiplicative chaos and applications: A review
In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already
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