Diffusion limited aggregation, resetting and large deviations of Brownian motion

Abstract

Models of fractal growth commonly consider particles diffusing in a medium and that stick irreversibly to the forming aggregate when making contact for the first time. As shown by the well-known diffusion limited aggregation (DLA) model and its generalisations, the fractal dimension is sensitive to the nature of the stochastic motion of the particles. Here, we study the structures formed by finite-lived Brownian particles, i.e. particles constrained to find the aggregate within a prescribed time, and which are removed otherwise. This motion can be modelled by diffusion with stochastic resetting, a class of processes which has been widely studied in recent years. In the short lifetime limit, a very small fraction of the particles manage to reach the aggregate. Hence, growth is controlled by atypical Brownian trajectories, that move nearly in straight line according to a large deviation principle. In d dimensions, the resulting fractal dimension of the aggregate decreases from the DLA value and tends to 1, instead of increasing to d as expected from ballistic aggregation. In the zero lifetime limit one recovers the non-trivial model of ‘aggregation by the tips’ proposed long ago by R. Jullien [J. Phys. A: Math. Gen. 19, 2129 (1986)].