The pleasures and pains of studying the two-type Richardson model

TL;DR

This survey reviews the two-type Richardson model, a stochastic process modeling competition on lattices, highlighting known results, open problems, and the conditions under which two competing entities can coexist infinitely.

Contribution

It provides a comprehensive overview of the two-type Richardson model, emphasizing the conjecture about coexistence depending on entity intensities and summarizing key known results and open questions.

Findings

Coexistence likely occurs if and only if entities have equal intensity.

The model extends the classical single-entity Richardson model to competition scenarios.

Open problems remain regarding the conditions for infinite coexistence.

Abstract

This paper provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on $\mathbb{Z}^d$. In its simplest formulation, the Richardson model describes the evolution of a single infectious entity on $\mathbb{Z}^d$, but more recently the dynamics have been extended to comprise two competing growing entities. For this version of the model, the main question is whether there is a positive probability for both entities to simultaneously grow to occupy infinite parts of the lattice, the conjecture being that the answer is yes if and only if the entities have the same intensity. In this paper attention focuses on the two-type model, but the most important results for the one-type version are also described.