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Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology

- Ewain Gwynne, Jason Miller
- Mathematics
- 2 August 2016

We prove that the uniform infinite half-plane quadrangulation (UIHPQ), with either general or simple boundary, equipped with its graph distance, its natural area measure, and the curve which traces… Expand

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

- Ewain Gwynne, Jason Miller, S. Sheffield
- MathematicsThe Annals of Probability
- 31 May 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,… Expand

Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense

- Ewain Gwynne, N. Holden, Xin Sun
- Mathematics
- 3 March 2016

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… Expand

Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk

- Ewain Gwynne, Jason Miller
- MathematicsAnnales de l'Institut Henri Poincaré, Probabilit…
- 18 January 2017

We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in… Expand

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

- Jian Ding, Ewain Gwynne
- MathematicsCommunications in Mathematical Physics
- 3 July 2018

We prove that for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , there is an exponent $$d_\gamma > 2$$ d γ > 2 , the “fractal dimension of $$\gamma $$ γ -Liouville quantum gravity (LQG)”, which describes… Expand

A distance exponent for Liouville quantum gravity

- Ewain Gwynne, N. Holden, Xin Sun
- Mathematics
- 3 June 2016

Let $$\gamma \in (0,2)$$γ∈(0,2) and let h be the random distribution on $$\mathbb C$$C which describes a $$\gamma $$γ-Liouville quantum gravity (LQG) cone. Also let $$\kappa = 16/\gamma ^2… Expand

An almost sure KPZ relation for SLE and Brownian motion

- Ewain Gwynne, N. Holden, Jason Miller
- MathematicsThe Annals of Probability
- 3 December 2015

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… Expand

Mating of trees for random planar maps and Liouville quantum gravity: a survey

- Ewain Gwynne, N. Holden, Xin Sun
- Mathematics
- 10 October 2019

We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The… Expand

A mating-of-trees approach for graph distances in random planar maps

- Ewain Gwynne, N. Holden, Xin Sun
- MathematicsProbability Theory and Related Fields
- 2 November 2017

We introduce a general technique for proving estimates for certain random planar maps which belong to the $$\gamma $$ γ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ γ… Expand

Scaling limits for the critical Fortuin–Kasteleyn model on a random planar map I: Cone times

- Ewain Gwynne, Cheng Mao, Xin Sun
- MathematicsAnnales de l'Institut Henri Poincaré, Probabilit…
- 2 February 2015

Sheffield (2011) introduced an inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. He showed that… Expand

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