Scott Sheffield

Tug-of-war with noise: A game-theoretic view of the

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 two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and the player who wins the toss chooses a vector v ∈ B(0, ǫ) to add to the game position, after which a random “noise vector” with mean zero and variance q p |v| 2 in each orthogonal direction is also added. The game ends when the game position reaches some y ∈ ∂Ω, and player I’s payoff is F (y). We show that (for sufficiently regular Ω) as ǫ tends to zero the functions uǫ converge uniformly to the unique p-harmonic extension of F . Using a modified game (in which ǫ gets smaller as the game position approaches ∂Ω), we prove similar statements for general bounded domains Ω and resolutive functions F . These games and their variants interpolate between the tug of war games studied by Peres, Schramm, Sheffield, and Wilson (p = ∞) and the motion-by-curvature games introduced by Spencer and studied by Kohn and Serfaty (p = 1). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about p-capacity and p-harmonic measure.

p

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), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.

-Laplacian

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.

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

A metric space X has Markov type 2, if for any reversible flnite-state Markov chain fZtg (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfles E(D 2) • K 2 tE(D 2) for some K = K(X) 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 < 1, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension deflned on all of Lp.

Harmonic explorer and its convergence to SLE4

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of logcorrelated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

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 properly regularized quantum area measure dμγ = ε γ2/2eγhε(z)dz, where dz is Lebesgue measure on D, γ is a real parameter, 0 ≤ γ < 2, and hε(z) denotes the mean value on the circle of radius ε centered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case γ > 2 is shown to be related to the quantum measure dμγ′ , γ′ < 2, by the fundamental duality γγ′ = 4.

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

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.

Duality and the Knizhnik-Polyakov-Zamolodchikov relation in Liouville quantum gravity.

Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk B_r of radius r=\sqrt{n/\pi}. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant C such that the following holds with probability one: B_{r - C \log r} \subset A(\pi r^2) \subset B_{r+ C \log r} for all sufficiently large r.

Random-Turn Hex and Other Selection Games

We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) h on ℝ d , defined up to a global additive constant.

Logarithmic Fluctuations for Internal DLA

We fix n and say a square in the two-dimensional grid indexed by (x, y) has color c if x + y ≡ c (mod n). A 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 height functions from the vertices of R to 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.