Harmonic measure of critical curves

Stochastic interface dynamics in the Hele-Shaw cell

A one-parametric stochastic regularized dynamics of the interface in the Hele-Shaw cell is introduced. The short-distance regularization suggested by the aggregation model stabilizes the growth by preventing the formation of cusps at the interface and makes the interface dynamics chaotic. The introduced stochastic growth process generates universal complex patterns with the well-developed fjords of oil separating the fingers of water. In a long time asymptotic, by coupling a conformal field theory to the stochastic growth process, we introduce a set of observables (the martingales), whose expectation values are constant in time. The martingales are closely connected to degenerate representations of the Virasoro algebra and can be written in terms of conformal correlation functions. A direct link between Laplacian growth and conformal Liouville field theory with the central charge c≥25 is proposed.

Universal Formula for the Mean First Passage Time in Planar Domains

We derive a general exact formula for the mean first passage time (MFPT) from a fixed point inside a planar domain to an escape region on its boundary. The underlying mixed Dirichlet-Neumann boundary value problem is conformally mapped onto the unit disk, solved exactly, and mapped back. The resulting formula for the MFPT is valid for an arbitrary space-dependent diffusion coefficient, while the leading logarithmic term is explicit, simple, and remarkably universal. In contrast to earlier works, we show that the natural small parameter of the problem is the harmonic measure of the escape region, not its perimeter. The conventional scaling of the MFPT with the area of the domain is altered when diffusing particles are released near the escape region. These findings change the current view of escape problems and related chemical or biochemical kinetics in complex, multiscale, porous or fractal domains, while the fundamental relation to the harmonic measure opens new ways of computing and interpreting MFPTs.

Harmonic measure for critical Potts clusters

We present a technique, which we call "etching," which we use to study the harmonic measure of Fortuin-Kasteleyn clusters in the Q-state Potts model for Q=1-4 . The harmonic measure is the probability distribution of random walkers diffusing onto the perimeter of a cluster. We use etching to study regions of clusters which are extremely unlikely to be hit by random walkers, having hitting probabilities down to 10-4600. We find good agreement between the theoretical predictions of Duplantier and our numerical results for the generalized dimension D(q) including regions of small and negative q .