The two-type Richardson model in the half-plane

TL;DR

This paper investigates the two-type Richardson model in the upper half-plane, proving that coexistence occurs with positive probability only when the two infection types have equal spreading intensities.

Contribution

It extends the understanding of the Richardson model to the half-plane, establishing a precise condition for coexistence based on infection intensities.

Findings

Coexistence occurs with positive probability only if infection intensities are equal.

Different intensities prevent infinite coexistence in the half-plane.

The result confirms the conjecture for the half-plane setting.

Abstract

The two-type Richardson model describes the growth of two competing infection types on the two or higher dimensional integer lattice. For types that spread with the same intensity, it is known that there is a positive probability for infinite coexistence, while for types with different intensities, it is conjectured that infinite coexistence is not possible. In this paper we study the two-type Richardson model in the upper half-plane $\mathbb{Z}\times\mathbb{Z}_+$, and prove that coexistence of two types starting on the horizontal axis has positive probability if and only if the types have the same intensity.