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...
Optimal Holder exponent for the SLE path (2009)
Johansson, Fredrik, Lawler, Gregory F.
We prove an upper bound on the optimal H\"older exponent for the chordal SLE path parameterized by capacity and thereby establish the optimal exponent as conjectured by J. Lind. We also give a new...
Two-sided SLE8=3 and the Infinite Self-Avoiding Polygon (2008)
Gregory F. Lawler, Joan R. Lind
Abstract In this paper we construct two-sided SLE8=3 and describe why it is a model of the infinite self-avoiding polygon. 1 Introduction The Schramm-Loewner evolution, SLE^, as introudced in [9], is...
Edward A. Bender, Gregory F. Lawler, Robin Pemantle, Herbert S. Wilf
Abstract. Let n = b1 + · · · + bk = b ′ 1 + · · · + b ′ k be a pair of compositions of n into k positive parts. We say this pair is irreducible if there is no positive j < k for which b1...
Séminaire Lotharingien de Combinatoire 52 (2004), Article B50h Irreducible compositions (2008)
Edward A. Bender, Gregory F. Lawler, Robin Pemantle, Herbert S. Wilf
and the first return to the origin of a random walk Abstract. Let n = b1 + · · · + bk = b ′ 1 + · · · + b ′ k
COMMUNICATIONS in PROBABILITY LOOP-ERASED WALKS INTERSECT INFINITELY OFTEN IN FOUR DIMENSIONS (2008)
In this short note we show that the paths two independent loop-erased random walks in four dimensions intersect infinitely often. We actually prove the stronger result that the cut-points of the two...
COMMUNICATIONS in PROBABILITY LOOP-ERASED WALKS INTERSECT INFINITELY OFTEN IN FOUR DIMENSIONS (2008)
In this short note we show that the paths two independent loop-erased random walks in four dimensions intersect infinitely often. We actually prove the stronger result that the cut-points of the two...
Schramm-Loewner Evolution (2007)
This is the first expository set of notes on SLE I have written since publishing a book two years ago [45]. That book covers material from a year-long class, so I cannot cover everything there....
Dimension and natural parametrization for SLE curves (2007)
Some possible definitions for the natural parametrization of SLE (Schramm-Loewner evolution) paths are proposed in terms of various limits. One of the definitions is used to give a new proof of the...
Difference operators arising from random walks with symmetric increments are studied. If the random walk is spatially homogeneous, then estimates of the first and second differences of harmonic...
The configurational measure on mutually avoiding SLE paths (2006)
Kozdron, Michael J., Lawler, Gregory F.
We define multiple chordal SLEs in a simply connected domain by considering a natural configurational measure on paths. We show how to construct these measures so that they are conformally covariant...
Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk (2005)
Kozdron, Michael J.; University Of Regina, Canada; Kozdron@math.uregina.ca, Lawler, Gregory F.; Cornell University, USA; Lawler@math.cornell.edu
We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z2 starting on the boundary exits A at another specified boundary point. The estimates are uniform...
Estimates of random walk exit probabilities and application to loop-erased random walk (2005)
Kozdron, Michael J., Lawler, Gregory F.
We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are...
The Beurling Estimate for a Class of Random Walks (2004)
Lawler, Gregory F; Cornell University; Lawler@math.cornell.edu, Limic, Vlada; University Of British Columbia; Limic@math.ubc.ca
An estimate of Beurling states that if K is a curve from 0 to the unit circle in the complex plane, then the probability that a Brownian motion starting at -&epsilon reaches the unit circle without...
Lawler, Gregory F., Ferreras, José A. Trujillo
The Brownian loop soup introduced in Lawler and Werner (2004) is a Poissonian realization from a sigma-finite measure on unrooted loops. This measure satisfies both conformal invariance and a...
Irreducible compositions and the first return to the origin of a random walk (2004)
Bender, Edward A., Lawler, Gregory F., Pemantle, Robin, Wilf, Herbert S.
Let $n = b_1 + ... + b_k = b_1' + \cdot + b_k'$ be a pair of compositions of $n$ into $k$ positive parts. We say this pair is {\em irreducible} if there is no positive $j < k$ for which $b_1 + ......
The Beurling estimate for a class of random walks (2004)
Lawler, Gregory F., Limic, Vlada
An estimate of Beurling states that if K is a curve from 0 to the unit circle in the complex plane, then the probability that a Brownian motion starting at -eps reaches the unit circle without...
Conformal invariance of planar loop-erased random walks and uniform spanning trees (2004)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $Dsubsetneqq\C$ is equal to the radial SLE$_2$ path. In particular, the limit exists and is...
Lawler, Gregory F., Werner, Wendelin
We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of...
On the scaling limit of planar self-avoiding walk (2002)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In...
Conformal invariance of planar loop-erased random walks and uniform spanning trees (2001)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It...
One-Arm Exponent for Critical 2D Percolation (2001)
Lawler, Gregory F.; Duke University And Cornell University; Lawler@math.cornell.edu, Schramm, Oded; Microsoft Research; Schramm@microsoft.com, Werner, Wendelin; Université Paris-Sud And IUF; Werner@math.u-psud.fr
The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than R is proved to decay like R to the power 5/48 as R goes to infinity.
One-arm exponent for critical 2D percolation (2001)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R^{-5/48}$ as $R\to\infty$.
The dimension of the planar Brownian frontier is 4=3 (2001)
Gregory F. Lawler, Oded Schramm, Wendelin Werner
The purpose of this note is to announce and sketch the proofs of results determining the Hausdorff dimension of certain subsets of planar Brownian paths. Proofs are currently written down in a...
The Dimension of the Planar Brownian Frontier is 4/3 (2000)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is...
Values of Brownian intersection exponents III: Two-sided exponents (2000)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
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...
Analyticity of intersection exponents for planar Brownian motion (2000)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let $B$ and $B'$ be independent planar Brownian motions started from distinct points, and define the...
Values of Brownian intersection exponents II: Plane exponents (2000)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
We derive the exact value of intersection exponents between planar Brownian motions or random walks, confirming predictions from theoretical physics by Duplantier and Kwon. Let B and B' be...
Strict Concavity of the Half Plane Intersection Exponent for Planar Brownian Motion (2000)
Lawler, Gregory F.; Duke University And Cornell University; Lawler@math.cornell.edu
The intersection exponents for planar Brownian motion measure the exponential decay of probabilities of nonintersection of paths. We study the intersection exponent $xi(lambda_1,lambda_2)$ for...
Values of Brownian intersection exponents I: Half-plane exponents (1999)
Lawler, Gregory F., Schramm, Oded, Werner, Wendelin
This paper proves conjectures originating in the physics literature regarding the intersection exponents of Brownian motion in a half-plane. For instance, suppose that B and B' are two independent...
Intersection Exponents for Planar Brownian Motion (1999)
Lawler, Gregory F., Werner, Wendelin
We derive properties concerning all intersection exponents for planar Brownian motion and we define generalized exponents that, loosely speaking, correspond to noninteger numbers of Brownian paths....
Cut Times for Brownian Motion and Random Walk (1999)
A cut time for a Brownian motion or a random walk is a time at which the past and the future of the process to not intersect. In this paper we review the work of Erdos on cut times and discuss more...
Loop-Erased Walks Intersect Infinitely Often in Four Dimensions (1998)
Lawler, Gregory F.; Duke University; Jose@math.duke.edu
In this short note we show that the paths two independent loop-erased random walks in four dimensions intersect infinitely often. We actually prove the stronger result that the cut-points of the two...
A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions (1998)
The growth exponent $\alpha$ for loop-erased or Laplacian random walk on the integer lattice is defined by saying that the expected time to reach the sphere of radius $n$ is of order $n^\alpha$. We...
Strict Concavity of the Intersection Exponent for Brownian Motion in Two and Three Dimensions (1998)
The intersection exponent for Brownian motion is a measure of how likely Brownian motion paths in two and three dimensions do not intersect. We consider the intersection exponent () = d (k; ) as a...
Intersection Exponents for Planar Brownian Motion (1998)
Gregory F. Lawler, Wendelin Werner
We derive properties concerning all intersection exponents for planar Brownian motion and we define generalized exponents that loosely speaking correspond to non-integer numbers of Brownian paths....
A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions (1998)
The growth exponent ff for loop-erased or Laplacian random walk on the integer lattice is defined by saying that the expected time to reach the sphere of radius n is of order n ff . We prove that in...
There is a close relationship between critical exponents for proabilities of events and fractal properties of paths of Brownian motion and random walk in two and three dimensions. Cone points, cut...
Loop-Erased Random Walk (1998)
this paper is to summarize some of the results about looperased walk and to give some of the important open questions. We will not try to cover everything about loop-erased random walk (LERW), but...
Multifractal Nature of Two Dimensional Simple Random Walk Paths (1997)
The multifractal spectrum of discrete harmonic measure of a two dimensional simple random walk path is considered. It is shown that the spectrum is the same as for Brownian motion, is nontrivial, and...
The Frontier of a Brownian Path Is Multifractal (1997)
We consider the multifractal spectrum of harmonic measure of a Brownian motion path in two or three dimensions. We show that the multifractal spectrum is nontrivial and relate the spectrum to the...
Cut Times for Simple Random Walk (1996)
Lawler, Gregory F.; Duke University And Cornell University; Lawler@math.cornell.edu
Let $S(n)$ be a simple random walk taking values in $Z^d$. A time $n$ is called a cut time if [ S[0,n] cap S[n+1,infty) = emptyset . ] We show that in three dimensions the number of cut times less...
The Dimension of the Frontier of Planar Brownian Motion (1996)
Lawler, Gregory F.; Duke University; Jose@math.duke.edu
Let $B$ be a two dimensional Brownian motion and let the frontier of $B[0,1]$ be defined as the set of all points in $B[0,1]$ that are in the closure of the unbounded connected component of its...
Hausdorff Dimension of Cut Points for Brownian Motion (1995)
Lawler, Gregory F.; Duke University And Cornell University; Lawler@math.cornell.edu
Let $B$ be a Brownian motion in $R^d$, $d=2,3$. A time $tin [0,1]$ is called a cut time for $B[0,1]$ if $B[0,t) cap B(t,1] = emptyset$. We show that the Hausdorff dimension of the set of cut times...
Cut Times For Simple Random Walk (1995)
: Let S(n) be a simple random walk taking values in Z d . A time n is called a cut time if S[0; n] " S[n + 1; 1) = ;: We show that in three dimensions the number of cut times less than n grows...
Nonintersecting Planar Brownian Motions (1995)
In this paper we construct a measure on pairs of Brownian motions starting at the same point conditioned so their paths do not intersect. The construction of this measure is a start towards the...
Cut times for simple random walk (1995)
Gregory F. Lawler, Gregory F. Lawler
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