# Competing frogs on $\mathbb{Z}^d$

## TL;DR

This paper studies a two-type frog model on  lattice, showing both types can activate infinitely many particles with positive probability and can coexist when their movement probabilities are equal.

## Contribution

It introduces a two-type frog model with random initial particles and analyzes conditions for both types to survive and coexist.

## Key findings

- Both types can activate infinitely many particles with positive probability.
- Coexistence occurs when movement probabilities are equal.
- Open problems include effects of heavy-tailed initial distributions.

## Abstract

A two-type version of the frog model on $\mathbb{Z}^d$ is formulated, where active type $i$ particles move according to lazy random walks with probability $p_i$ of jumping in each time step ($i=1,2$). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles at one other site are activated and assigned type 2, while all other particles are sleeping. When an active type $i$ particle moves to a new site, any sleeping particles there are activated and assigned type $i$, with an arbitrary tie-breaker deciding the type if the site is hit by particles of both types in the same time step. We show that the event $G_i$ that type $i$ activates infinitely many particles has positive probability for all $p_1,p_2\in(0,1]$ ($i=1,2$). Furthermore, if $p_1=p_2$, then the types can coexist in the sense that $\mathbb{P}(G_1\cap G_2)>0$. We also formulate several open problems. For instance, we conjecture that, when the initial number of particles per site has a heavy tail, the types can coexist also when $p_1\neq p_2$.