Growth by Random Walker Sampling, and Scaling of the Dielectric Breakdown Model

TL;DR

This paper introduces a novel sampling method using random walker recurrence times to estimate nonlinear moments of the Harmonic Measure, enabling efficient simulation and analysis of complex growth models like DLA and Dielectric Breakdown.

Contribution

It presents a new technique based on recurrence times for simulating Dielectric Breakdown Model growth and revises earlier exponent estimates, challenging previous superuniversality claims.

Findings

Recurrence times accurately probe multifractal growth measures.

The new method efficiently simulates large clusters in the Dielectric Breakdown Model.

Previous conformal mapping results and superuniversality hypotheses are refuted.

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.