First Passage Percolation with Recovery

• Published in: arXiv.org 2024-10
• Main Authors: Candellero, Elisabetta, Garcia-Sanchez, Tom
• Format: Article
• Language: English
• Subjects: Apexes; Clocks; Percolation; Seeds
• ISSN: 2331-8422

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 \(\limsup_{t}\frac{M_t}{t}\leq \varepsilon \).