Accessibility Percolation with Rough Mount Fuji labels
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910220560629760 |
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| author | Bellon, Diana De Armas Roberts, Matthew I. |
| author_facet | Bellon, Diana De Armas Roberts, Matthew I. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29561 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Accessibility Percolation with Rough Mount Fuji labels Bellon, Diana De Armas Roberts, Matthew I. Probability 60K35 (Primary) 60J80, 92D15 (Secondary) 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. |
| title | Accessibility Percolation with Rough Mount Fuji labels |
| topic | Probability 60K35 (Primary) 60J80, 92D15 (Secondary) |
| url | https://arxiv.org/abs/2603.29561 |