# A new upper bound for the critical probability of the frog model on homogeneous trees

## TL;DR

This paper establishes a new, improved upper bound for the critical probability in the frog model on homogeneous trees, providing a closed-form formula and confirming a longstanding conjecture.

## Contribution

It introduces a tighter upper bound for the critical survival probability of the frog model on homogeneous trees, advancing previous theoretical results.

## Key findings

- Derived a new upper bound for the critical probability
- Provided a closed-form formula for the upper bound
- Confirmed a conjecture from prior research

## Abstract

We consider the interacting particle system on the homogeneous tree of degree $(d + 1)$, known as frog model. In this model, active particles perform independent random walks, awakening all sleeping particles they encounter, and dying after a random number of jumps, with geometric distribution. We prove an upper bound for the critical parameter of survival of the model, which improves the previously known results. This upper bound was conjectured in a paper by Lebensztayn et al. ($ J. Stat. Phys.$, 119(1-2), 331-345, 2005). We also give a closed formula for the upper bound.