arXiv subjects

Scott Sheffield

Publications and source records attributed to Scott Sheffield.

88 records · Page 5Linked to original sources

arXiv2008

Contour lines of the two-dimensional discrete Gaussian free field

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 discrete Gaussian free field on a Jordan domain -- with boundary values -lambda on one boundary arc and lambda on the complementary arc -- the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are -a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4;a/lambda-1,b/lambda-1), a variant of SLE(4).

math.PR↗

arXiv2008

Tug-of-war and the infinity Laplacian

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 all open subsets U of X that do not intersect Y. This was previously known only for bounded domains R^n, in which case u is infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also prove the first general uniqueness results for Delta_infty u = g on bounded subsets of R^n (when g is uniformly continuous and bounded away from zero), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be the value of the following two-player zero-sum game, called tug-of-war: fix x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y, and player one's payoff is F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i) We show that the u^εconverge uniformly to u as epsilon tends to zero. Even for bounded domains in R^n, the game theoretic description of infinity-harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity-harmonic functions in the unit disk with boundary values supported in a delta-neighborhood of a Cantor set on the unit circle.

math.AP↗

arXiv2007

Hausdorff dimension of the SLE curve intersected with the real line

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 consequence we are able to prove that the SLE curve intersected with the real line has Hausdorff dimension 2-8/kappa, almost surely.

math.PR↗

arXiv2006

Gaussian free fields for mathematicians

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 simple random walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm-Loewner evolution.

math.PR↗

arXiv2006

Exploration trees and conformal loop ensembles

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 are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties. We conjecture that they are the scaling limits of various random loop models from statistical physics, including the O(n) loop models.

math.PR↗

arXiv2006

Random Surfaces

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 by a full-rank sublattice L of Z^d; they include many discrete and continuous height models (e.g., domino tilings, square ice, the harmonic crystal, the Ginzburg-Landau grad-phi interface model, the linear solid-on-solid model) as special cases. A gradient phase is an L-ergodic gradient Gibbs measure with finite specific free energy. A gradient phase is smooth if it is the gradient of an ordinary Gibbs measure; otherwise it is rough. We prove a variational principle--characterizing gradient phases of a given slope as minimizers of the specific free energy--and an empirical measure large deviations principle (with a unique rate function minimizer) for random surfaces on mesh approximations of bounded domains. Using a geometric technique called "cluster swapping" (a variant of the Swendsen-Wang update for Fortuin-Kasteleyn clusters), we also prove that the surface tension is strictly convex and that if u is in the interior of the space of finite-surface tension slopes, then there exists a minimal energy gradient phase mu_u of slope u. This mu_u is always unique for real valued random surfaces. In the discrete models, mu_u is unique if at least one of the following holds: d is in {1, 2}, there exists a rough gradient phase of slope u, or u is irrational. When d=2, the slopes of all smooth phases (a.k.a. crystal facets) lie in the dual lattice of L.

math.PR↗

arXiv2006

Uniqueness of maximal entropy measure on essential spanning forests

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 of $G$ for which $\bigcup G_n=G$. Pemantle's arguments imply that the uniform measures on spanning trees of $G_n$ converge weakly to an $\operatorname {Aut}(G)$-invariant measure $μ_G$ on essential spanning forests of $G$. We show that if $G$ is a connected, amenable graph and $Γ\subset \operatorname {Aut}(G)$ acts quasitransitively on $G$, then $μ_G$ is the unique $Γ$-invariant measure on essential spanning forests of $G$ for which the specific entropy is maximal. This result originated with Burton and Pemantle, who gave a short but incorrect proof in the case $Γ\cong\mathbb{Z}^d$. Lyons discovered the error and asked about the more general statement that we prove.

math.PR↗

arXiv2006

Random-Turn Hex and other selection games

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 edges, while White wins by completing a crossing between the other pair of opposite edges. Although ordinary Hex is famously difficult to analyze, Random-Turn Hex--in which players toss a coin before each turn to decide who gets to place the next stone--has a simple optimal strategy. It belongs to a general class of random-turn games--called selection games--in which the expected payoff when both players play the random-turn game optimally is the same as when both players play randomly. We also describe the optimal strategy and study the expected length of the game under optimal play for Random-Turn Hex and several other selection games.

math.PR↗

arXiv2006

The harmonic explorer and its convergence to SLE(4)

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 of the path from a point near the end of the current path. We prove that the harmonic explorer converges to SLE(4) as the grid gets finer.

math.PR↗

arXiv2004

Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces

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 to $X$, the distance $D_t$ from $f(Z_0)$ to $f(Z_t)$ satisfies $\E(D_t^2) \le K^2 t \E(D_1^2)$ for some $K=K(X)<\infty$. This notion is due to K. Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, only Hilbert space (and its bi-Lipschitz equivalents) were known to have Markov type 2. We show that every Banach space with modulus of smoothness of power type 2 (in particular, $L_p$ for $p>2$) has Markov type 2; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for $1<q<2<p<\infty$, any Lipschitz mapping from a subset of $L_p$ to $L_q$ has a Lipschitz extension defined on all of $L_p$.

math.FA↗

arXiv2003

Linear speed large deviations for percolation clusters

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, endowed with the Hausdorff metric delta. When d >= 2, and Gamma is any open subset of Omega, we prove: lim_{n \to \infty}(1/n) \log P(C_n \in Γ) = -\inf_{S \in Γ} λ(S) where lambda(S) is the one-dimensional Hausdorff measure of S defined using the {\em correlation norm}: ||u|| := \lim_{n \to \infty} - \frac{1}{n} \log P (u_n \in C_n) where u_n is u rounded to the nearest element of (1/n)Z^d. Given points a^1, >..., a^k in R^d, there are finitely many correlation-norm Steiner trees spanning these points and the origin. We show that if the C_n are each conditioned to contain the points a^1_n,..., a^k_n, then the probability that C_n fails to approximate one of these trees decays exponentially in n.

math.PR↗

arXiv2003

Dimers and Amoebae

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 explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. Further, we prove that the spectral curve of a dimer model is always a real curve of special type, namely it is a Harnack curve. This implies many qualitative and quantitative statement about the behavior of the dimer model, such as existence of smooth phases, decay rate of correlations, growth rate of height function fluctuations, etc.

math-ph↗

arXiv2003

Dimers, Tilings and Trees

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; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.

math.CO↗

arXiv2003

A large-deviation theorem for tree-indexed Markov chains

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 vertex. We prove a large deviation principle in $n$, with explicit rate function, for the empirical subtree measures of multitype Galton-Watson trees conditioned to have exactly $n$ vertices. In the process, we extend the notions of shift-invariance and specific relative entropy--as typically understood for Markov fields on deterministic graphs such as $\mathbb Z^d$--to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with $n$ vertices.

math.PR↗

arXiv2001

Ribbon Tilings and Multidimensional Height Functions

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 square of each color. We show that the set of order-$n$ ribbon tilings of a simply connected region $R$ is in one-to-one correspondence with a set of {\it height functions} from the vertices of $R$ to $\mathbb Z^{n}$ satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph. Using these facts, we describe a linear (in the area of $R$) algorithm for determining whether $R$ can be tiled with ribbon tiles of order $n$ and producing such a tiling when one exists. We also resolve a conjecture of Pak by showing that any pair of order-$n$ ribbon tilings of $R$ can be connected by a sequence of local replacement moves. Some of our results are generalizations of known results for order-2 ribbon tilings (a.k.a. domino tilings). We also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems.

math.CO↗

arXiv2001

Computing and Sampling Restricted Vertex Degree Subgraphs and Hamiltonian Cycles

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 a single face---and if each subgraph contains an alternating set of the edges of that face. For a given $ϕ: V \mapsto \mathbb Z^+$, $S_ϕ$ is the set of subgraphs of $G$ in which each $v\in V$ has degree $ϕ(v)$. Two elements of $S_ϕ$ are said to be adjacent if they differ by a $Z$-transformation. We determine the connected components of $S_ϕ$ and assign a {\it height function} to each of its elements. If $ϕ$ is identically two, and $G$ is a grid graph, $S_ϕ$ contains the partitions of the vertices of $G$ into cycles. We prove that we can always apply a series of $Z$-transformations to decrease the total number of cycles provided there is enough ``slack'' in the corresponding height function. This allows us to determine in polynomial time the minimal number of cycles into which $G$ can be partitioned provided $G$ has a limited number of non-square faces. In particular, we determine the Hamiltonicity of polyomino graphs in $O(|V|^2)$ steps. The algorithm extends to $n$-holed-torus-embedded graphs that have grid-like properties. We also provide Markov chains for sampling and approximately counting the Hamiltonian cycles of $G$.

math.CO↗