For the frog model with discrete Weibull lifetimes having right-edge density exponent β and lifetime exponent γ, the process dies out a.s. when β > 1/γ (finite-mean initial configuration) and survives with positive probability when β < 1/γ (non-empty initial configuration).
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The Frog Model on $\mathbb Z$ with Random Discrete Weibull Lifetimes and Biased Nearest-Neighbour Random Walks
For the frog model with discrete Weibull lifetimes having right-edge density exponent β and lifetime exponent γ, the process dies out a.s. when β > 1/γ (finite-mean initial configuration) and survives with positive probability when β < 1/γ (non-empty initial configuration).