Integrability-preserving regularizations of Laplacian Growth

TL;DR

This paper explores methods to regularize Laplacian Growth models while preserving their integrability, aiming to better understand their long-term scaling behavior and relation to Diffusion-Limited Aggregation.

Contribution

It proposes a new approach for integrability-preserving regularizations of Laplacian Growth, connecting it to DLA and addressing long-standing scaling property challenges.

Findings

Outline of a potential theory for regularized LG

Discussion of the relation between LG and DLA

Identification of challenges in describing long-time scaling

Abstract

The Laplacian Growth (LG) model is known as a universality class of scale-free aggregation models in two dimensions, characterized by classical integrability and featuring finite-time boundary singularity formation. A discrete counterpart, Diffusion-Limited Aggregation (or DLA), has a similar local growth law, but significantly different global behavior. For both LG and DLA, a proper description for the scaling properties of long-time solutions is not available yet. In this note, we outline a possible approach towards finding the correct theory yielding a regularized LG and its relation to DLA.