arXiv subjects

Fredrik Viklund

Publications and source records attributed to Fredrik Viklund.

24 records · Page 2Linked to original sources

arXiv2019

Remarks on the regularity of quasislits

A quasislit is the image of a vertical line segment [0, iy], y > 0, under a quasiconformal homeomorphism of the upper half-plane fixing infinity. Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-1/2 class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-1/2 seminorm, s. In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the full range s <4. The estimate is shown to be sharp for small s. As consequences, we derive explicit Hölder exponents for s < 4 as well as estimates on winding rates. We also relate quantitatively the Lip-1/2 seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.

math.CV↗

arXiv2019

Schramm's formula and the Green's function for multiple SLE

We construct martingale observables for systems of multiple SLE curves by applying screening techniques within the CFT framework recently developed by Kang and Makarov, extended to admit multiple SLEs. We illustrate this approach by rigorously establishing explicit expressions for the Green's function and Schramm's formula in the case of two curves growing towards infinity. In the special case when the curves are `fused' and start at the same point, some of the formulas we prove were predicted in the physics literature.

math.PR↗

arXiv2019

One-dimensional scaling limits in a planar Laplacian random growth model

We consider a family of growth models defined using conformal maps in which the local growth rate is determined by $|Φ_n'|^{-η}$, where $Φ_n$ is the aggregate map for $n$ particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits. More precisely, we exhibit a phase transition in the ancestral structure of the growing clusters: for $η>1$, aggregating particles attach to their immediate predecessors with high probability, while for $η<1$ almost surely this does not happen.

math.PR↗

arXiv2017

The Loewner difference equation and convergence of loop-erased random walk

We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. This coupling is different than the ones previously considered in this context. Our recent work (arXiv:1603.05203) on the convergence of LERW parametrized by length to SLE(2) parameterized by Minkowski content uses specific features of the coupling constructed here.

math.PR↗

arXiv2017

Convergence of radial loop-erased random walk in the natural parametrization

In recent work we have shown that loop-erased random walk (LERW) connecting two boundary points of a domain converges to the chordal Schramm-Loewner evolution (SLE(2)) in the sense of curves parametrized by Minkowski content. In this note we explain how to derive the analogous result for LERW from a boundary point to an interior point, converging towards radial SLE(2).

math.PR↗

arXiv2016

Convergence of loop-erased random walk in the natural parametrization

Loop-erased random walk, abbreviated LERW, is one of the most well-studied critical lattice models. It is the self-avoiding random walk one gets after erasing the loops from a simple random walk in order or alternatively by considering the branches in a uniformly chosen spanning tree. This paper proves that planar LERW parametrized by renormalized length converges in the lattice size scaling limit to SLE(2) parametrized by 5/4-dimensional Minkowski content. In doing this we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theorem, several of our results about LERW are of independent interest: for example, two-point estimates, estimates on maximal content, and a "separation lemma".

math.PR↗