arXiv subjects
Publications and source records attributed to Gregory F. Lawler.
The Green's function for the chordal Schramm-Loewner evolution $SLE_κ$ for $0 < κ< 8$, gives the normalized probability of getting near points. We give up-to-constant bounds for the two-point Green's function.
We give estimates for the probability that a chordal, radial or two-sided radial SLE$_κ$ curve retreats far from its terminal point after coming close to it, for $κ\leq 4$. The estimates are uniform over all initial segments of the curve, and are sharp up to a universal constant.
Loop measures and their associated loop soups are generally viewed as arising from finite state Markov chains. We generalize several results to loop measures arising from potentially complex edge weights. We discuss two applications: Wilson's algorithm to produce uniform spanning trees and an isomorphism theorem due to Le Jan.
The Brownian loop measure is a conformally invariant measure on loops in the plane that arises when studying the Schramm-Loewner evolution (SLE). When an SLE curve in a domain evolves from an interior point, it is natural to consider the loops that hit the curve and leave the domain, but their measure is infinite. We show that there is a related normalized quantity that is finite and invariant under Möbius transformations of the plane. We estimate this quantity when the curve is small and the domain simply connected. We then use this estimate to prove a formula for the Radon-Nikodym derivative of reversed radial SLE with respect to whole-plane SLE.
In this paper we define and prove of the existence of the multi-point Green's function for SLE - a normalized limit of the probability that an $SLE_κ$ curve passes near to a pair of marked points in the interior of a domain. When $κ<8$ this probability is nontrivial, and an expression can be written in terms two-sided radial SLE. One of the main components to our proof is a refinement of a bound first provided by Beffara [Ann. Probab. 36 (2008) 1421-1452]. This work contains a proof of this bound independent from the original.
Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_κ$ curves is well defined for all $κ<8$. Our proof uses a two-interior-point local martingale.
We give a new proof of a result of Rick Kenyon that the probability that an edge in the middle of an n x n square is used in a loop-erased walk connecting opposites sides is of order n^{-3/4}. We, in fact, improve the result by showing that this estimate is correct up to multiplicative constants.
We consider pairs of 3-dimensional Brownian paths, started at the origin and conditioned to have no intersections after time zero. We show that there exists a unique measure on pairs of paths that is invariant under this conditioning, while improving the previously known rate of convergence to stationarity.
The natural paramterization or length for the Schramm-Loewner evolution (SLEκ) is the candidate for the scaling limit of the length of discrete curves for κ< 8. We improve the proof of the existence of the parametrization and use this to establish some new results. In particular, we show that the natural parametrization is independent of domain and it is Hölder continuous with respect to the capacity parametrization. We also give up-to-constants bounds for the two-point Green's function. Although we do not prove the conjecture that the natural length is given by the appropriate Minkowski content, we do prove that the corresponding expectations converge.
We prove the existence of the Green's function for radial SLE(k) for k<8. Unlike the chordal case where an explicit formula for the Green's function is known for all values of k<8, we give an explicit formula only for k=4. For other values of k, we give a formula in terms of an expectation with respect to SLE conditioned to go through a point.
We consider the two-dimensional self-avoiding walk (SAW) in a simply connected domain that contains the origin. The SAW starts at the origin and ends somewhere on the boundary. The distribution of the endpoint along the boundary is expected to differ from the SLE partition function prediction for this distribution because of lattice effects that persist in the scaling limit. We give a precise conjecture for how to compute this lattice effect correction and support our conjecture with simulations. We also give a precise conjecture for the lattice corrections that persist in the scaling limit of the lambda-SAW walk.
We define the Schramm-Loewner evolution (SLE) in multiply connected domains for kappa \leq 4 using the Brownian loop measure. We show that in the case of the annulus, this is the same measure obtained recently by Dapeng Zhan. We use the loop formulation to give a different derivation of the partial differential equation for the partition function for the annulus.
We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as $β\rightarrow β_c-$ of the probability measure on all finite length walks $ω$ with the probability of $ω$ proportional to $β_c^{|ω|}$ where $|ω|$ is the number of steps in $ω$. The self-avoiding walk in a strip $\{z : 0<\Im(z)<y\}$ is defined by considering all self-avoiding walks $ω$ in the strip which start at the origin and end somewhere on the top boundary with probability proportional to $β_c^{|ω|}$ We prove that this probability measure may be obtained by conditioning the SAW in the half plane to have a bridge at height $y$. This observation is the basis for simulations to test conjectures on the distribution of the endpoint of the SAW in a strip and the relationship between the distribution of this strip SAW and SLE$_{8/3}$.
The tip multifractal spectrum of a two-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm-Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability one, and we give applications to the scaling of harmonic measure at the tip.
We prove that radial $SLE_κ$ and two-sided radial $SLE_κ$ are continuous at their terminal point.
The Schramm-Loewner evolution (SLE_κ) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ< 8, an instance of SLE_κis a random planar curve with almost sure Hausdorff dimension d = 1 + κ/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume. For κ< 8, we use a Doob-Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE_κthat transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is non-trivial (i.e., the curve is not entirely traversed in zero time) for κ< 4(7 - \sqrt{33}) = 5.021 ....
We prove an upper bound on the optimal Hölder 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 proof of the lower bound. Our proofs are based on the sharp estimates of moments of the derivative of the inverse map. In particular, we improve an estimate of the second author.
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. However, these notes are not just a subset of those notes, because there is a slight change of perspective. The main differences are: o I have defined SLE as a finite measure on paths that is not necessarily a probability measure. This seems more natural from the perspective of limits of lattice systems and seems to be more useful when extending SLE to non-simply connected domains. (However, I do not discuss non-simply connected domains in these notes.) o I have made more use of the Girsanov theorem in studying corresponding martingales and local martingales. As in [45], I will focus these notes on the continuous process SLE and will not prove any results about convergence of discrete processes to SLE. However, my first lecture will be about discrete processes -- it is very hard to appreciate SLE if one does not understand what it is trying to model.