The Frog Model on $\mathbb{Z}$ with Discrete Weibull Lifetimes and Random Parameter $p$

TL;DR

This paper analyzes extinction–survival phase transitions for a one-dimensional frog process on $\mathbb{Z}$ with random survival parameter $\pi$ and discrete Weibull lifetimes, where conditional lifetimes satisfy $P(\Xi\ge k|\pi=p)=p^{k^{\gamma}}$. By deriving precise one-particle displacement tails and leveraging regular variation and stable-subordinator representations, the authors establish a sharp $\gamma$-dependent threshold $\beta_c=\frac{1}{2\gamma}$ that separates almost-sure extinction from survival with positive probability, with boundary behavior dictated by the slowly varying function $L$. The results recover the geometric case ($\gamma=1$) with $\beta_c=\tfrac{1}{2}$ and provide explicit boundary criteria, thereby enriching the understanding of aging lifetimes and random environments in frog-like activation models. The techniques blend Laplace-transform methods, Tauberian arguments, and stable-process representations to connect aging, edge behavior, and random initial configurations into a cohesive phase diagram on the integer lattice.

Abstract

We study the frog model on $\mathbb{Z}$ with particle wise discrete Weibull lifetimes. Each particle has an i.i.d. survival parameter $π\in(0,1)$; conditionally on $π=p$, its lifetime $Ξ$ satisfies \[ P(Ξ\ge k\mid π=p)=p^{k^γ},\qquad k\in\mathbb{N}_0,γ>0. \] The law of $π$ has right edge density \[ f_π(u)\sim(1-u)^{β-1},L\big((1-u)^{-1}\big)\qquad (u\uparrow 1), \] with $β>0$ and $L$ slowly varying; let $η$ denote the common law of the i.i.d. initial occupation numbers $\{η_x\}_{x\in\mathbb{Z}}$. The survival parameter distribution strictly extends the Beta family, while the lifetime distribution extends the geometric case. We prove a sharp extinction and survival dichotomy with the $γ-$dependent threshold \[ β_c:=\frac{1}{2γ}. \] If $β>β_c$ and $E(η)<\infty$, the process becomes extinct almost surely; if $β<β_c$ and $P(η=0)<1$, it survives with positive probability. At the boundary $β=β_c$ we provide explicit criteria in terms of $\limsup/\liminf$ of $L(n^{2γ})$. The case $γ=1$ (geometric lifetimes) recovers the benchmark $β_c=\frac{1}{2}$ and the critical refinements previously obtained for random geometric lifetimes.

Figures (2)

• Figure 1: Survival tails of the discrete Weibull, $\mathbb{P}(\Xi\ge k)=p^{k^{\gamma}}$, for several values of $p$ and $\gamma$. For $\gamma>1$ the tail decays faster than geometric (wear--out); $\gamma=1$ recovers the geometric (memoryless) case; and $0<\gamma<1$ yields heavier tails (burn--in).
• Figure 2: Phase diagram in the $(\beta,\gamma)$ plane. The critical curve $\gamma_c(\beta)=1/(2\beta)$ separates survival with positive probability (below) from extinction.

Theorems & Definitions (27)

• Theorem 1.1: General extinction–survival dichotomy
• Corollary 1.2: One-sided envelope criteria
• Remark 1.3
• Remark 1.4
• Proposition 2.1: Upper bound $\gamma > 1$
• Proposition 2.2: Lower bound $\gamma>1$
• Theorem 2.3: Case $\gamma>1$
• Theorem 2.4: Case $\gamma<1$
• proof : Proof of Proposition \ref{['prop:upper-gamma>1-EN-eps']}
• proof : Proof of Proposition \ref{['prop:lower-gamma>1-EN-eps']}