Coexistence in discrete time Multi-type competing Frog Models

TL;DR

This paper investigates conditions under which multiple types of particles in discrete-time frog models on integer lattices can coexist, demonstrating positive probability coexistence for various configurations and types.

Contribution

It extends coexistence results to multi-type frog models on b^d, including multiple and infinite coexistence scenarios with rich initial configurations.

Findings

Positive probability coexistence for two types on b^d with any jumping parameters.

Existence of coexistence among 2^d types for sufficiently rich initial configurations.

Infinite coexistence possible on b^d for d  3 with rich initial setups.

Abstract

We study coexistence in discrete time multi-type frog models. We first show that for two types of particles on $\mathbb{Z}^d$, for $d\geq2$, for any jumping parameters $p_1, p_2 \in (0,1)$, coexistence occurs with positive probability for sufficiently rich deterministic initial configuration. We extend this to the case of random distribution of initial particles. We study the question of coexistence for multiple types and show positive probability coexistence of $2^d$ types on $\mathbb{Z}^d$ for rich enough initial configuration. We also show an instance of infinite coexistence on $\mathbb{Z}^d$ for $d \geq 3$ provided we have sufficiently rich initial configuration.