First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph $G$ place a red particle at a reference vertex $o$ and colorless particles (seeds) at all other vertices. The red particle starts spreading a \emph{ red first passage percolation} of rate $1$, while all seeds are dormant. As soon as a seed is reached by the process, it turns red and starts spreading { red first passage percolation}. All vertices are equipped with independent exponential clocks ringing at rate $\gamma>0$, when a clock rings the corresponding \emph{ red vertex turns black}. For $t\geq 0$, let $H_t$ and $M_t$ denote the size of the longest red path and of the largest red cluster present at time $t$. %, respectively. If $G$ is the semi-line, then for all $\gamma>0$ almost surely $\limsup_{ t}\frac{ H_t\log\log t}{ \log t}=1 $ and $\liminf_{ t}H_t=0$. In contrast, if $G$ is an infinite Galton-Watson tree with offspring mean $\mathbf{ m}>1$ then, for all $\gamma>0$, almost surely $\liminf_{ t}\frac{ H_t\log t}{ t}\geq\mathbf{ m}-1 $ and $\liminf_{ t}\frac{ M_t\log\log t}{ t}\geq \mathbf{ m}-1$, while $\limsup_{ t} \frac{ M_t}{ e^{ c t}}\leq 1$, for all $c>\mathbf{ m} -1$. Also, almost surely as $t\to \infty$, for all $\gamma>0$ $H_t$ is of order at most $t$. Furthermore, if we restrict our attention to bounded-degree graphs, then for any $\varepsilon>0$ there is a critical value $\gamma_c>0$ so that for all $\gamma>\gamma_c$, almost surely $\limsup_{ t}\frac{ M_t}{ t}\leq \varepsilon $.
Categories: math.PR