61 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
Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines
The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner
driving term. It is finite if and only if the curve is a Weil-Petersson
quasicircle. In this paper, we describe cutting and welding operations on
finite Dirichlet energy functions defined in the plane, allowing expression of
the Loewner energy in terms of Dirichlet energy dissipation. We show that the
Loewner energy of a unit vector field flow-line is equal to the Dirichlet
energy of the harmonically extended winding. We also give an identity involving
a complex-valued function of finite Dirichlet energy that expresses the welding
and flow-line identities simultaneously. As applications, we prove that
arclength isometric welding of two domains is sub-additive in the energy, and
that the energy of equipotentials in a simply connected domain is monotone. Our
main identities can be viewed as action functional analogs of both the welding
and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian
free field.Comment: 28 pages, 3 figures. Minor revision according to referees' repor
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
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