Growth by random walker sampling and scaling of the dielectric breakdown model

Abstract

Random walkers absorbing on a boundary sample the Harmonic Measure linearly and independently: we discuss how the recurrence times between impacts enable non-linear moments of the measure to be estimated. From this we derive a new technique to simulate Dielectric Breakdown Model growth which is governed nonlinearly by the Harmonic Measure. Recurrence times are shown to be accurate and effective in probing the multifractal growth measure of diffusion limited aggregation. For the Dielectric Breakdown Model our new technique grows large clusters efficiently and we are led to significantly revise earlier exponent estimates. Previous results by two conformal mapping techniques were less converged than expected, and in particular a recent theoretical suggestion of superuniversality is firmly refuted.

Random walks, diffusion limited aggregation in a wedge, and average conformal maps

“…We have shown [24] that this estimate is adequate to allow us to grow DBM clusters. In our work here we use a 3 to estimate the probability.…”

“…For each member of the ensemble there is a point, r(θ) whose image is a point on the unit circle at e iθ . Our definition of the ensemble average shape generated by the DLA or DBM process [24] is the ensemble average of r(θ), i.e., the centroid of those points. In Figures (3) and (4), below we show the average shapes in 90…”

“…We also change the prefactor, A, as we go along to make an efficient code. For details, see [24]. Examples of this is shown in Fig.…”

“…This algorithm is very slow, and is not practical for generating large clusters. Recently [24] we have introduced a method of growing DBM clusters by random walker sampling. The key to this method is to define the age, a 1 of a growth site.…”

Anisotropic diffusion limited aggregation in three dimensions: Universality and nonuniversality

“…The Diffusion Limited Aggregation (DLA) [1] model has been the focus of a great deal of research due both to the fractal [2,3,4] and multifractal [5,6,7] properties of the clusters it produces, and to its underlying mathematical connection to diverse problems including solidification [8,9], viscous fingering [10] and electrodeposition [11,12]. Its key feature is that the surface irreversibly absorbs an incident diffusive flux, and growth velocity is locally proportional to that flux density.…”