The Frog Model on $\mathbb{Z}$ with General Random Survival Parameter

Paper

Abstract

We study the frog model on

with particle-wise random geometric lifetimes: each particle has a survival parameter

sampled i.i.d., whose density near

satisfies

with

, and

slowly varying. This strictly extends the

case. Let

denote the common law of the i.i.d.\ initial number of particles

. Using a percolation comparison and sharp one-particle displacement tails, we obtain a universal threshold at

. If

and

, extinction occurs almost surely. If

and

, survival has positive probability. At the boundary

we give sharp criteria: extinction if

and

; survival if

and

. These results recover the Carvalho-Machado threshold for Beta laws and show that only the exponent

governs the phase transition, while

impacts the critical regime.