Fredrik Johansson Viklund

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.

Convergence rates for loop-erased random walk and other Loewner curves

We estimate convergence rates for curves generated by Loewners 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 Warschawskis 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 SLE2 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 Holder continuous in the capacity parameterization, assuming its driving term is Holder continuous. We also briefly discuss the case when the curves are a priori known to be Holder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.

A Dimension Spectrum for SLE Boundary Collisions

We consider chordal SLE

Vacuum-driven lipase-catalysed direct condensation of L-ascorbic acid and fatty acids in ionic liquids: synthesis of a natural surface active antioxidant

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Almost sure multifractal spectrum for the tip of an SLE curve

κ curves for

Ionic liquids create new opportunities for nonaqueous biocatalysis with polar substrates: Acylation of glucose and ascorbic acid

{\kappa > 4}