On Hele-Shaw problems arising as scaling limits

TL;DR

This paper investigates the scaling limits of discrete 2D aggregation models related to Hele-Shaw flows, analyzing their properties and providing exact formulas for their shapes using conformal mappings.

Contribution

It introduces conjectural scaling limits for these models and derives explicit shape formulas, advancing understanding of Hele-Shaw flow behaviors in discrete aggregation.

Findings

Predicted exact shape formulas for aggregation models

Analyzed moment properties of Hele-Shaw solutions

Applied conformal mappings to solve shape problems

Abstract

We discuss conjectural scaling limits of discrete 2-dimensional aggregation models conditioned on a semi-axis considered by Levine and Peres in arXiv:0712.3378. These are certain problems about Hele-Show flows. We study moment properties of their solutions, and solve some of them using conformal mappings. In particular, we predict the exact formula for the computer-generated shape on the left side of Fig. 4 in arXiv:0712.3378.