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Fractional Gaussian fields: A survey

- A. Lodhia, S. Sheffield, Xin Sun, Samuel S. Watson
- Mathematics, Physics
- 21 July 2014

We discuss a family of random fields indexed by a parameter s ∈ R which we call the fractional Gaussian fields, given by FGFs(R) = (−∆)−s/2W, where W is a white noise on Rd and (−∆)−s/2 is the… Expand

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… Expand

Two Perspectives of the 2D Unit Area Quantum Sphere and Their Equivalence

- Juhan Aru, Yichao Huang, Xin Sun
- Physics
- 19 December 2015

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… 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

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

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

On Fluctuations for Random Band Toeplitz Matrices

In this paper we study two one-parameter families of random band Toeplitz matrices: \[ A_n(t)=\frac{1}{\sqrt{b_n}}\Big(a_{i-j}\delta_{|i-j|\le[b_nt]}\Big)_{i,j=1}^n \quad\text{and}\quad… Expand

Equivalence of Liouville measure and Gaussian free field

- N. Berestycki, S. Sheffield, Xin Sun
- Mathematics
- 20 October 2014

Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $\gamma \in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity… Expand

Schnyder woods, SLE$_{(16)}$, and Liouville quantum gravity

- Yiting Li, Xin Sun, Samuel S. Watson
- Mathematics
- 9 May 2017

In 1990, Schnyder used a 3-spanning-tree decomposition of a simple triangulation, now known as the Schnyder wood, to give a fundamental grid-embedding algorithm for planar maps. We show that a… 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

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