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## Dimers and amoebae

- R. Kenyon, A. Okounkov, S. Sheffield
- Mathematics
- 5 November 2003

We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive… Expand

## Liouville quantum gravity and KPZ

- B. Duplantier, S. Sheffield
- Mathematics
- 11 August 2008

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… Expand

## Tug-of-war and the infinity Laplacian

- Y. Peres, O. Schramm, S. Sheffield, D. Wilson
- Mathematics
- 28 April 2006

We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a… Expand

## Conformal weldings of random surfaces: SLE and the quantum gravity zipper

- S. Sheffield
- Mathematics
- 21 December 2010

We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE),… Expand

## Conformal loop ensembles: the Markovian characterization and the loop-soup construction

- S. Sheffield, W. Werner
- Mathematics
- 11 June 2010

For random collections of self-avoiding loops in two-dimensional domains, we dene a simple and natural conformal restriction property that is conjecturally satised by the scaling limits of interfaces… Expand

## Tug-of-war with noise: A game-theoretic view of the $p$-Laplacian

- Y. Peres, S. Sheffield
- Mathematics
- 29 July 2006

Fix a bounded domain Ω ⊂ Rd, a continuous function F : ∂Ω → R, and constants ǫ > 0 and 1 < p, q < ∞ with p−1 + q−1 = 1. For each x ∈ Ω, let uǫ(x) be the value for player I of the following… Expand

## Imaginary geometry I: interacting SLEs

- Jason Miller, S. Sheffield
- Mathematics
- 6 January 2012

Fix constants $$\chi >0$$χ>0 and $$\theta \in [0,2\pi )$$θ∈[0,2π), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field $$e^{i(h/\chi… Expand

## Exploration trees and conformal loop ensembles

- S. Sheffield
- Mathematics
- 6 September 2006

We construct and study the conformal loop ensembles CLE(kappa), defined for all kappa between 8/3 and 8, using branching variants of SLE(kappa) called exploration trees. The conformal loop ensembles… Expand

## Gaussian free fields for mathematicians

- S. Sheffield
- Mathematics
- 4 December 2003

The d-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the… Expand

## Liouville quantum gravity as a mating of trees

- B. Duplantier, Jason Miller, S. Sheffield
- Physics
- 24 September 2014

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… Expand

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