The initial set in the frog model is irrelevant

TL;DR

This paper proves that in the frog model on integer lattices, the long-term behavior and coexistence possibilities are independent of the initial activation set or configuration, highlighting a form of universality.

Contribution

It establishes that the asymptotic shape and coexistence conditions in the frog model are unaffected by initial configurations, extending understanding of the model's robustness.

Findings

Asymptotic shape is independent of initial set in one-type model.

Coexistence probability does not depend on initial configuration in two-type model.

Results demonstrate universality in the frog model's long-term behavior.

Abstract

In this note, we consider the frog model on $\mathbb{Z}^d$ and a two-type version of it with two types of particles. For the one-type model, we show that the asymptotic shape does not depend on the initially activated set and the configuration there. For the two-type model, we show that the possibility for the types to coexist in that both of them activate infinitely many particles does not depend on the choice of the initially activated sets and the configurations there.