The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

@article{Gwynne2017TheTE,
  title={The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity},
  author={Ewain Gwynne and Jason Miller and Scott Sheffield},
  journal={The Annals of Probability},
  year={2017}
}
We prove that the Tutte embeddings (a.k.a. harmonic/embeddings) of certain random planar maps converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and $\gamma$ ranges from $0$ to $2$ as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLE$_{\kappa}$ for $\kappa =16/\gamma^2$ (in the annealed… 

Figures from this paper

Random walks on mated-CRT planar maps and Liouville Brownian motion
We prove a scaling limit result for random walk on certain random planar maps with its natural time parametrization. In particular, we show that for $\gamma \in (0,2)$, the random walk on the
Harmonic functions on mated-CRT maps
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar
Convergence of uniform triangulations under the Cardy embedding
We consider an embedding of planar maps into an equilateral triangle $\Delta$ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation
An almost sure KPZ relation for SLE and Brownian motion
The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $\gamma$-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a
Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense
Recent works have shown that random triangulations decorated by critical ($p=1/2$) Bernoulli site percolation converge in the scaling limit to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface
The Tutte Embedding of the Poisson–Voronoi Tessellation of the Brownian Disk Converges to $$\sqrt{8/3}$$-Liouville Quantum Gravity
Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a
Volume of metric balls in Liouville quantum gravity
We study the volume of metric balls in Liouville quantum gravity (LQG). For $\gamma \in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of
Weak LQG metrics and Liouville first passage percolation
For $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we define a weak $$\gamma $$ γ - Liouville quantum gravity ( LQG ) metric to be a function $$h\mapsto D_h$$ h ↦ D h which takes in an instance of the planar
LIOUVILLE QUANTUM GRAVITY AS A METRIC SPACE AND A SCALING LIMIT
  • Jason Miller
  • Mathematics, Physics
    Proceedings of the International Congress of Mathematicians (ICM 2018)
  • 2019
Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal
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
...
...

References

SHOWING 1-10 OF 147 REFERENCES
Harmonic functions on mated-CRT maps
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar
Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense
Kenyon, Miller, Sheffield, and Wilson (2015) showed how to encode a random bipolar-oriented planar map by means of a random walk with a certain step size distribution. Using this encoding together
Liouville Brownian motion
We construct a stochastic process, called the Liouville Brownian motion which we conjecture to be the scaling limit of random walks on large planar maps which are embedded in the euclidean plane or
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
An almost sure KPZ relation for SLE and Brownian motion
The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $\gamma$-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a
Brownian motion correlation in the peanosphere for $\kappa > 8$
The peanosphere (or "mating of trees") construction of Duplantier, Miller, and Sheffield encodes certain types of $\gamma$-Liouville quantum gravity (LQG) surfaces ($\gamma \in (0,2)$) decorated with
Liouville quantum gravity spheres as matings of finite-diameter trees
We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure
SLE as a Mating of Trees in Euclidean Geometry
The mating of trees approach to Schramm–Loewner evolution (SLE) in the random geometry of Liouville quantum gravity (LQG) has been recently developed by Duplantier et al. (Liouville quantum gravity
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
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
...
...