Random-turn Hex and other selection games (2009)
Yuval Peres, Oded Schramm, Scott Sheffield, David B. Wilson
Overview. The game of Hex, invented independently by Piet Hein in 1942 and John Nash in 1948 [9], has two players who take turns placing stones of their respective colors on the hexagons of a...
The natural parametrization for the Schramm-Loewner evolution (2009)
Lawler, Gregory F., Sheffield, Scott
The Schramm-Loewner evolution (SLE_\kappa) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When \kappa < 8, an instance of SLE_\kappa is a random...
Fixation for Distributed Clustering Processes (2009)
Hilario, Marcelo R., Louidor, Oren, Newman, Charles M., Rolla, Leonardo T., Sheffield, Scott, Sidoravicius, Vladas
We study a discrete-time resource flow in $Z^d$, where wealthier vertices attract the resources of their less rich neighbors. For any translation-invariant probability distribution of initial...
Power law Polya's urn and fractional Brownian motion (2009)
Hammond, Alan, Sheffield, Scott
We introduce a natural family of random walks on the set of integers that scale to fractional Brownian motion. The increments X_n have the property that given {X_k: k < n}, the conditional law of X_n...
Duality and KPZ in Liouville Quantum Gravity (2009)
Duplantier, Bertrand, Sheffield, Scott
We present a (mathematically rigorous) probabilistic and geometrical proof of the KPZ relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the...
Conformal radii for conformal loop ensembles (2009)
Oded Schramm, Scott Sheffield, David B. Wilson
The conformal loop ensembles CLEκ, defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the...
The Covariant Measure of SLE on the Boundary (2008)
Alberts, Tom, Sheffield, Scott
We construct a natural measure mu supported on the intersection of a chordal SLE(kappa) curve gamma with the real line R, in the range 4 < kappa < 8. The measure is a function of the SLE path in...
Liouville Quantum Gravity and KPZ (2008)
Duplantier, Bertrand, Sheffield, Scott
Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure...
Hausdorff Dimension of the SLE Curve Intersected with the Real Line (2008)
Alberts, Tom; Courant Institute Of Mathematical Sciences; Alberts@cims.nyu.edu, Sheffield, Scott; Courant Institute Of Mathematical Sciences; Sheff@cims.nyu.edu
We establish an upper bound on the asymptotic probability of an SLE(kappa) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < kappa < 8. As a...
Ribbon tilings and multidimensional height function, preprint arXiv:math.CO/0107095 (2008)
Abstract. We fix n and say a square in the two-dimensional grid indexed by (x, y)hascolorc if x + y ≡ c (mod n). A ribbon tile of order n is a connected polyomino containing exactly one square of...
TUG-OF-WAR AND THE INFINITY LAPLACIAN (2008)
Yuval Peres, Oded Schramm, Scott Sheffield, B. Wilson
1.1. Overview. 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...
Hausdorff dimension of the SLE curve intersected with the real line (2007)
Alberts, Tom, Sheffield, Scott
We establish an upper bound on the asymptotic probability of an SLE(kappa) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < kappa < 8. As a...
Conformal radii for conformal loop ensembles (2006)
Schramm, Oded, Sheffield, Scott, Wilson, David B.
The conformal loop ensembles CLE(k), defined for k in [8/3, 8], are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the...
Exploration trees and conformal loop ensembles (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...
Tug of war with noise: a game theoretic view of the p-Laplacian (2006)
Peres, Yuval, Sheffield, Scott
Fix a bounded domain Omega in R^d, a continuous function F on the boundary of Omega, and constants epsilon>0, p>1, and q>1 with p^{-1} + q^{-1} = 1. For each x in Omega, let u^epsilon(x) be the value...
Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces (2006)
Naor, Assaf, Peres, Yuval, Schramm, Oded, Sheffield, Scott
A metric space $X$ has Markov-type $2$ if for any reversible finite-state Markov chain $\{Z_t\}$ (with $Z_0$ chosen according to the stationary distribution) and any map $f$ from the state space to...
Contour lines of the two-dimensional discrete Gaussian free field (2006)
Schramm, Oded, Sheffield, Scott
We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant lambda > 0 such that when h is an interpolation of the...
Uniqueness of maximal entropy measure on essential spanning forests (2006)
An essential spanning forest of an infinite graph G is a spanning forest of G in which all trees have infinitely many vertices. Let Gn be an increasing sequence of finite connected subgraphs of G for...
Kenyon, Richard, Okounkov, Andrei, Sheffield, Scott
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...
Tug-of-war and the infinity Laplacian (2006)
Peres, Yuval, Schramm, Oded, Sheffield, Scott, Wilson, David B.
We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u for which Lip_U u = Lip_{boundary of U} u for...
Kenyon, Richard, Sheffield, Scott, Okounkov, Andrei
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...
Harmonic explorer and its convergence to SLE4 (2005)
Schramm, Oded, Sheffield, Scott
The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left-hand side...
Random-Turn Hex and other selection games (2005)
Peres, Yuval, Schramm, Oded, Sheffield, Scott, Wilson, David B.
The game of Hex has two players who take turns placing stones of their respective colors on the hexagons of a rhombus-shaped hexagonal grid. Black wins by completing a crossing between two opposite...
Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces (2004)
Naor, Assaf, Peres, Yuval, Schramm, Oded, Sheffield, Scott
A metric space $X$ has {\em Markov type} 2, if for any reversible finite-state Markov chain $\{Z_t\}$ (with $Z_0$ chosen according to the stationary distribution) and any map $f$ from the state space...
Uniqueness of maximal entropy measure on essential spanning forests (2004)
An essential spanning forest of an infinite graph $G$ is a spanning forest of $G$ in which all trees have infinitely many vertices. Let $G_n$ be an increasing sequence of finite connected subgraphs...
Gaussian free fields for mathematicians (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...
Kenyon, Richard, Okounkov, Andrei, Sheffield, Scott
We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on an weighted, bipartite, doubly periodic graph G embedded in the plane. We derive...
Linear speed large deviations for percolation clusters (2003)
Kovchegov, Yevgeniy, Sheffield, Scott
Let C_n be the origin-containing cluster in subcritical percolation on the lattice (1/n) Z^d, viewed as a random variable in the space Omega of compact, connected, origin-containing subsets of R^d,...
The harmonic explorer and its convergence to SLE(4) (2003)
Schramm, Oded, Sheffield, Scott
The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left-hand side...
Dimers, Tilings and Trees (2003)
Kenyon, Richard, Sheffield, Scott
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains;...
A large-deviation theorem for tree-indexed Markov chains (2003)
Dembo, Amir, Morters, Peter, Sheffield, Scott
Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed by taking all descendants of a single...
We study "random surfaces," which are random real (or integer) valued functions on Z^d. The laws are determined by convex, nearest neighbor, difference potentials that are invariant under translation...
Random surfaces : large deviations principles and gradient Gibbs measure classifications / (2003)
Sheffield, Scott., Dembo, Amir Advisor
Submitted to the Department of Mathematics.
Ribbon Tilings and Multidimensional Height Functions (2001)
We fix $n$ and say a square in the two-dimensional grid indexed by $(x,y)$ has color $c$ if $x+y \equiv c \pmod{n}$. A {\it ribbon tile} of order $n$ is a connected polyomino containing exactly one...
Computing and Sampling Restricted Vertex Degree Subgraphs and Hamiltonian Cycles (2000)
Let $G=(V,E)$ be a bipartite graph embedded in a plane (or $n$-holed torus). Two subgraphs of $G$ differ by a {\it $Z$-transformation} if their symmetric difference consists of the boundary edges of...