2025/03/12 by de Carvalho, Gustavo O., Machado, Fábio P.
#05C81 #60K35 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2503.09766
We study the frog model on \( ℤ \) with geometric lifetimes, introducing a random survival parameter. Active and inactive particles are placed at the vertices of \( ℤ \). The lifetime of each active particle follows a geometric random variable with parameter \( 1-p \), where \( p \) is randomly sampled from a distribution \( π\). Each active particle performs a simple random walk on \( ℤ \) until it dies, activating any inactive particles it encounters along its path. In contrast to the usual case where \( p \) is fixed, we show that there exist non-trivial distributions \( π\) for which the model survives with positive probability. More specifically, for π∼ Beta(α,β), we establish the existence of a critical value \( β=0.5 \), that separates almost sure extinction from survival with positive probability. Furthermore, we show that the model is recurrent whenever it survives with positive probability.