Scaling limit of DLA on a long line segment

TL;DR

This paper proves that the large-scale behavior of DLA on a long line segment converges to a stationary process, overcoming multi-scale interaction challenges through coupling and intermediate processes.

Contribution

It introduces a coupling scheme and an intermediate DLA process to establish the scaling limit of DLA on a line segment, addressing complex multi-scale interactions.

Findings

DLA on a long line segment converges to SDLA in the scaling limit

Coupling scheme effectively handles multi-scale interactions

Intermediate process with absorbing boundaries facilitates analysis

Abstract

In this paper, we prove that the bulk of DLA starting from a long line segment on the $x$-axis has a scaling limit to the stationary DLA process (SDLA). The main phenomenological difficulty is the multi-scale, non-monotone interaction of the DLA arms. We overcome this via a coupling scheme between the two processes and an intermediate DLA process with absorbing mesoscopic boundary segments.