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$_2$ 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 $D$ be a Jordan domain of unit capacity. We study the partition function of a planar Coulomb gas in $D$ with a hard wall along $\eta = \partial D$, $$Z_{n}(D) =\frac 1{n!}\int_{D^n}\prod_{1\le k < \ell \le n}|z_k-z_\ell|^{2} \prod_{k=1}^n d^2z_k.$$ We are interested in how the geometry of $\eta$ is reflected in the large $n$ behavior of $Z_n(D)$. We prove that $\eta$ is a Weil-Petersson quasicircle if and only if $$\log Z_n(D)= \log Z_n(\mathbb{D}) -I^L(\eta)/12 + o(1), \quad n\to \infty,$$ where $I^L(\eta)$ is the Loewner energy of $\eta$, $\mathbb{D}$ is the unit disc, and $\log Z_n(\mathbb{D}) = \log n!/\pi^n$. We next consider piecewise analytic $\eta$ with $m$ corners of interior opening angles $\pi \alpha_p, p=1,\ldots, m$. Our main result is the asymptotic formula $$\log Z_n(D)= \log Z_n(\mathbb{D}) - \frac 16\sum_{p=1}^m \left(\alpha_p+\frac 1{\alpha_p}-2 \right) \log n + o(\log n), \quad n\to \infty,$$ which is consistent with physics predictions. The starting point of our analysis is an exact expression for $\log Z_{n}(D)$ in terms of a Fredholm determinant involving the truncated Grunsky operator for $D$. 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