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First passage percolation in hostile environment is not monotone

Elisabetta Candellero, Alexandre Stauffer

Abstract

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.

First passage percolation in hostile environment is not monotone

Abstract

We study a natural growth process with competition, modeled by two first passage percolation processes, and , spreading on a graph. starts at the origin and spreads at rate , whereas starts from a random set of \emph{inactive seeds} distributed as Bernoulli percolation of parameter . A seed of gets activated when one of the two processes attempts to occupy its location, and from this moment onwards spreads at some fixed rate . In previous works~[17, 3, 7] it has been shown that when both or are small enough, then \emph{survives} (i.e., it occupies an infinite set of vertices) with positive probability. It might seem intuitive that decreasing or is beneficial to . However, we prove that, in general, this is indeed false by constructing a graph for which the probability that survives is not a monotone function of or , implying the existence of multiple phase transitions. This behavior contrasts with other natural growth processes such as the -type Richardson model.

Paper Structure

This paper contains 26 sections, 19 theorems, 205 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Idea of the proof
  3. Construction of the graph.
  4. Analysis of FPPHE for $\mu=\mu_1$ and $\mu=\mu_2$.
  5. Conclusion of the proof.
  6. Organization of the paper.
  7. Passage times
  8. Construction of the graph
  9. The tile $G$: construction, notation and some facts
  10. Behavior of FPPHE on a tile
  11. The infinite graph $\mathbb{G}^\infty$
  12. Proof of Theorem \ref{['thm:main1']}
  13. Analysis of the process on the tile
  14. Proof of Lemma \ref{['lemma:toProp-mu1']}
  15. Proof of Lemma \ref{['lemma:toProp-mu1']}(i), "lower part" of $G$.
  16. ...and 11 more sections

Key Result

theorem 1

There is a connected, infinite graph of bounded degree and values $\mu_1,\mu_2$ with $0<\mu_1<\mu_2<1$, such that, whenever the rate $\lambda$ is small enough,

Figures (6)

  • Figure 1: Sketch of $\overline{\mathbb{T}}_d^h$: the triangle on the left (pink) represents the tree $\mathbb{T}_d^h$, while the one on the right (black) represents the set of edges connecting the $h$-th generation of $\mathbb{T}_d^h$ to $W$.
  • Figure 2: Sketch of the tile $G$ and the lengths of its components.
  • Figure 3: The tile $G$ that gives non-monotonicity in $\lambda$ as in Remark \ref{['rem:main2']} part \ref{['it:main2']}.
  • Figure 4: The construction of $\mathbb{G}^\infty$, every diamond represents a copy of the tile $G$, while arrows inside each tile point from the origin of the tile to the tail. Considering vertex $v$ as the origin of $\mathbb{G}^\infty$, we obtain that $\vec{\mathbb{G}}^\infty$ (as formally defined in the proof of Theorems \ref{['thm:main1']} and \ref{['thm:main2']}) as the part of the figure to the right of $v$.
  • Figure 5: Sketch of the construction of $h(\mathcal{N}_j)$.
  • ...and 1 more figures

Theorems & Definitions (48)

  • theorem 1
  • remark 1
  • theorem \oldthetheorem
  • definition 1: First Passage Times
  • remark 2
  • lemma 1
  • proposition 1
  • remark 3
  • proposition 2
  • theorem \oldthetheorem
  • ...and 38 more