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Accessibility Percolation with Rough Mount Fuji labels

Diana De Armas Bellon, Matthew I. Roberts

Abstract

Consider an infinite, rooted, connected graph where each vertex is labelled with an independent and identically distributed Uniform(0,1) random variable, plus a parameter $θ$ times its distance from the root $ρ$. That is, we label vertex $v$ with $X_v = U_v + θd(ρ,v)$. We say that accessibility percolation occurs if there is an infinite path started from $ρ$ along which the vertex labels are increasing. When the graph is a Bienaymé-Galton-Watson tree, we give an exact characterisation of the critical value $θ_c$ such that there is accessibility percolation with positive probability if and only if $θ>θ_c$. We also give more explicit bounds on the value of $θ_c$. The lower bound holds for a much more general class of trees. When the graph is the lattice $\mathbb{Z}^n$ for $n\ge 2$, we show that there is a non-trivial phase transition and give some first bounds on $θ_c$. To do this we introduce a novel coupling with oriented percolation.

Accessibility Percolation with Rough Mount Fuji labels

Abstract

Consider an infinite, rooted, connected graph where each vertex is labelled with an independent and identically distributed Uniform(0,1) random variable, plus a parameter times its distance from the root . That is, we label vertex with . We say that accessibility percolation occurs if there is an infinite path started from along which the vertex labels are increasing. When the graph is a Bienaymé-Galton-Watson tree, we give an exact characterisation of the critical value such that there is accessibility percolation with positive probability if and only if . We also give more explicit bounds on the value of . The lower bound holds for a much more general class of trees. When the graph is the lattice for , we show that there is a non-trivial phase transition and give some first bounds on . To do this we introduce a novel coupling with oriented percolation.

Paper Structure

This paper contains 20 sections, 22 theorems, 121 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Results on trees
  3. Results on $\mathbb{Z}^n$
  4. Motivation
  5. Existing and related work
  6. Layout of article
  7. Upper bound on the probability of one path being increasing
  8. There is no Rough Mount Fuji accessibility percolation on trees for small $\theta$
  9. The case of Bienaymé-Galton-Watson trees: Proof of Proposition \ref{['prop:no_perco_GW']}
  10. The case of general trees
  11. Proof of Theorem \ref{['theomultitype']}: characterising the critical value using a multitype branching process approach
  12. The theorem of Biggins and Kyprianou
  13. An expression for the eigenfunctions of the increasing reproduction operator
  14. Warm-up: proof of survival in Theorem \ref{['theomultitype']} for $\theta\in(1/2,1)$
  15. Survival in Theorem \ref{['theomultitype']} in the general case $\theta\in(0,1)$
  16. ...and 5 more sections

Key Result

Theorem 1.2

For $\theta\in(0,1]$ and $x\in\mathbb{R}$, define Let $m_c(\theta)$ be the minimal root of the polynomial equation Suppose that $T$ is a Bienaymé-Galton-Watson tree whose offspring distribution $L$ satisfies $\mathbb{E}[L]=m$ and $\mathbb{E}[L\log_+ L] < \infty$. There is Rough Mount Fuji accessibility percolation with positive probability on $T$ if and only if $m > m_c(\theta)$.

Figures (4)

  • Figure 1: Accessible paths from the origin with backsteps. The left-hand image uses the $\ell^1$ distance, and shows several values of $\theta$, from $\theta=0.33$ (yellow) to $0.42$ (red). The middle image uses the $\ell^2$ distance, and shows $\theta=0.45$ (yellow) to $0.54$ (red).The right-hand image uses the $\ell^4$ distance, and shows $\theta=0.48$ (yellow) to $0.57$ (red). With the $\ell^1$ distance, paths near the diagonal are preferred. With the $\ell^4$ distance, paths near the axes are preferred. With the $\ell^2$ distance, there does not appear to be a strong directional preference.
  • Figure 2: Accessible paths from the origin without backsteps. The left-hand image uses the $\ell^1$ distance, and shows several values of $\theta$, from $\theta=0.33$ (yellow) to $0.42$ (red). The middle image uses the $\ell^2$ distance, and shows $\theta=0.45$ (yellow) to $0.54$ (red). The right-hand image uses the $\ell^4$ distance, and shows $\theta=0.53$ (yellow) to $0.62$ (red). In all cases, paths near the axes are inaccessible, but as $q$ increases, paths near the diagonal become more difficult too.
  • Figure 3: One brick, with the horizontal edges drawn in red, the left-vertical edges drawn in blue, and the right-vertical edges drawn in purple.
  • Figure 4: A small portion of the bricklayer lattice with each vertex replaced by a brick. Only open edges are drawn. Good bricks are coloured green and bad bricks are coloured red. A directed path of good bricks guarantees a path of open edges from the bottom-left of the first brick to the middle-right vertex of the last brick, both highlighted in yellow.

Theorems & Definitions (49)

  • Definition 1.1: Rough Mount Fuji
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 39 more