21 research outputs found
Convergence rates for loop-erased random walk and other Loewner curves
We estimate convergence rates for curves generated by Loewner's differential
equation under the basic assumption that a convergence rate for the driving
terms is known. An important tool is what we call the tip structure modulus, a
geometric measure of regularity for Loewner curves parameterized by capacity.
It is analogous to Warschawski's boundary structure modulus and closely related
to annuli crossings. The main application we have in mind is that of a random
discrete-model curve approaching a Schramm-Loewner evolution (SLE) curve in the
lattice size scaling limit. We carry out the approach in the case of
loop-erased random walk (LERW) in a simply connected domain. Under mild
assumptions of boundary regularity, we obtain an explicit power-law rate for
the convergence of the LERW path toward the radial SLE path in the supremum
norm, the curves being parameterized by capacity. On the deterministic side, we
show that the tip structure modulus gives a sufficient geometric condition for
a Loewner curve to be H\"{o}lder continuous in the capacity parameterization,
assuming its driving term is H\"{o}lder continuous. We also briefly discuss the
case when the curves are a priori known to be H\"{o}lder continuous in the
capacity parameterization and we obtain a power-law convergence rate depending
only on the regularity of the curves.Comment: Published in at http://dx.doi.org/10.1214/13-AOP872 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Almost sure multifractal spectrum for the tip of an SLE curve
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.Comment: 43 pages, 2 figure
Coulomb gas and the Grunsky operator on a Jordan domain with corners
Let be a Jordan domain of unit capacity. We study the partition function
of a planar Coulomb gas in with a hard wall along ,
We are interested in how the geometry of is
reflected in the large behavior of . We prove that is a
Weil-Petersson quasicircle if and only if
where is the Loewner energy of , is the unit
disc, and . We next consider piecewise
analytic with corners of interior opening angles . Our main result is the asymptotic formula
which is consistent with physics predictions. The starting point of our
analysis is an exact expression for in terms of a Fredholm
determinant involving the truncated Grunsky operator for . The proof of the
main result is based on careful asymptotic analysis of the Grunsky
coefficients. As further applications of our method we also study the Loewner
energy and the related Fekete-Pommerenke energy, a quantity appearing in the
analysis of Fekete points, for equipotentials approximating the boundary of a
domain with corners. We formulate several conjectures and open problems.Comment: 50 pages, 2 figure
Scaling limits of anisotropic Hastings-Levitov clusters.
We consider a variation of the standard Hastings-Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations around the deterministic limit flow