First passage percolation in hostile environment is not monotone
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| Format: | Preprint |
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2021
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| author | Candellero, Elisabetta Stauffer, Alexandre |
| author_facet | Candellero, Elisabetta Stauffer, Alexandre |
| contents | We study a natural growth process with competition, modeled by two first passage percolation processes, $FPP_1$ and $FPP_λ$, spreading on a graph. $FPP_1$ starts at the origin and spreads at rate $1$, whereas $FPP_λ$ starts from a random set of \emph{inactive seeds} distributed as Bernoulli percolation of parameter $μ\in (0,1)$. A seed of $FPP_λ$ gets activated when one of the two processes attempts to occupy its location, and from this moment onwards spreads at some fixed rate $λ>0$. In previous works~[17, 3, 7] it has been shown that when both $μ$ or $λ$ are small enough, then $FPP_1$ \emph{survives} (i.e., it occupies an infinite set of vertices) with positive probability. It might seem intuitive that decreasing $μ$ or $λ$ is beneficial to $FPP_1$. However, we prove that, in general, this is indeed false by constructing a graph for which the probability that $FPP_1$ survives is not a monotone function of $μ$ or $λ$, implying the existence of multiple phase transitions. This behavior contrasts with other natural growth processes such as the $2$-type Richardson model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_05821 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | First passage percolation in hostile environment is not monotone Candellero, Elisabetta Stauffer, Alexandre Probability We study a natural growth process with competition, modeled by two first passage percolation processes, $FPP_1$ and $FPP_λ$, spreading on a graph. $FPP_1$ starts at the origin and spreads at rate $1$, whereas $FPP_λ$ starts from a random set of \emph{inactive seeds} distributed as Bernoulli percolation of parameter $μ\in (0,1)$. A seed of $FPP_λ$ gets activated when one of the two processes attempts to occupy its location, and from this moment onwards spreads at some fixed rate $λ>0$. In previous works~[17, 3, 7] it has been shown that when both $μ$ or $λ$ are small enough, then $FPP_1$ \emph{survives} (i.e., it occupies an infinite set of vertices) with positive probability. It might seem intuitive that decreasing $μ$ or $λ$ is beneficial to $FPP_1$. However, we prove that, in general, this is indeed false by constructing a graph for which the probability that $FPP_1$ survives is not a monotone function of $μ$ or $λ$, implying the existence of multiple phase transitions. This behavior contrasts with other natural growth processes such as the $2$-type Richardson model. |
| title | First passage percolation in hostile environment is not monotone |
| topic | Probability |
| url | https://arxiv.org/abs/2110.05821 |