Gregory F. Lawler

Values of Brownian intersection exponents, I: Half-plane exponents

Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., [4], [6]). For instance, by making the assumption that critical planar percolation behaves in a conformally invariant way in the scaling limit, and using ideas involving conformal field theory, Cardy [7] produced an exact formula for the limit, as N → ∞, of the probability that, in two-dimensional critical percolation, there exists a cluster crossing the rectangle [0, aN] × [0, bN]. Also, Duplantier and Saleur [13] predicted the “fractal dimension” of the hull of a very large percolation cluster. These are just two examples among many such predictions.

Bounds on the ² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality

We prove a general version of Cheegers inequality for discrete-time Markov chains and continuous-time Markovian jump processes, both reversible and nonreversible, with general state space. We also prove a version of Cheegers inequality for Markov chains and processes with killing. As an application, we prove L 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

Conformal restriction: The chordal case

We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane Η, say) which satisfy the conformal restriction” property, i.e., K connects two fixed boundary points (0 and ∞, say) and the law of K conditioned to remain in a simply connected open subset H of H is identical to that of Φ( K ), where Φ is a conformal map from H onto H with Φ(0) = 0 and Φ(∞) = ∞. The construction of this family relies on the stochastic Loewner evolution processes with parameter κ ≤ 8/3 and on their distortion under conformal maps. We show in particular that SLE 8/3 is the only random simple curve satisfying conformal restriction and relate it to the outer boundaries of planar Brownian motion and SLE 6 . See link below.

Values of Brownian intersection exponents III: Two-sided exponents

Abstract This paper determines values of intersection exponents between packs of planar Brownian motions in the half-plane and in the plane that were not derived in our first two papers. For instance, it is proven that the exponent ξ(3,3) describing the asymptotic decay of the probability of non-intersection between two packs of three independent planar Brownian motions each is (73−2 73 )/12 . More generally, the values of ξ(w1,…,wk) and ξ (w 1 ′,…,w k ′) are determined for all k⩾2, w1,w2⩾1, w3,…,wk∈[0,∞) and all w′1,…,w′k∈[0,∞). The proof relies on the results derived in our first two papers and applies the same general methods. We first find the two-sided exponents for the stochastic Loewner evolution processes in a half-plane, from which the Brownian intersection exponents are determined via a universality argument.

Weak convergence of a random walk in a random environment

Let πi(x),i=1,...,d,x∈Zd, satisfy πi(x)≧α>0, and π1(x)+...+πd(x)=1. Define a Markov chain onZd by specifying that a particle atx takes a jump of +1 in theith direction with probability 1/2πi(x) and a jump of −1 in theith direction with probability 1/2πi(x). If the πi(x) are chosen from a stationary, ergodic distribution, then for almost all π the corresponding chain converges weakly to a Brownian motion.

Random walk loop soup

The Brownian loop soup introduced by Lawler and Werner (2004) is a Poissonian realization from a σ-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlos, Major, and Tusnady. To make the paper self-contained, we include a proof of the approximation result that we need.

The probability of intersection of independent random walks in four dimensions

LetS1 andS2 be independent simple random walks of lengthn inZ4 starting at 0 andx0 respectively. If |x0|2≈n, it is shown that the probability that the paths intersect is of order (logn)−1. Ifx0=0, it is shown that the probability of no intersection of the paths decays no faster than (logn)−1 and no slower than (logn)−1/2. It is conjectured that (logn)−1/2 is the actual decay rate.

Analyticity of intersection exponents for planar Brownian motion

The goal of the present paper is to show that the intersection exponents for planar Brownian motions are analytic. Let k~>l be a positive integer and let X 1 , . . . , X k be independent Brownian motions in the complex plane C started from 0. Let Y, y1, y2, ... denote other independent planar Brownian motions started from 1, and let St be the random variable (measurable with respect to X 1, ..., X k)

Loop-Erased Random Walk

Loop-erased random walk (LERW) is a process obtained from erasing loops from simple random walk. This paper reviews some of the results and conjectures about LERW. In particular, we discuss the critical exponents for LERW, Wilson’s algorithm for generating uniform spanning trees with LERW, and the role of conformal invariance in studying LERW in two dimensions.

SLE curves and natural parametrization

Developing the theory of two-sided radial and chordal