Accessibility Percolation with Rough Mount Fuji labels

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 $\theta$ times its distance from the root $\rho$. That is, we label vertex $v$ with $X_v = U_v + \theta d(\rho,v)$. We say that accessibility percolation occurs if there is an infinite path started from $\rho$ along which the vertex labels are increasing. When the graph is a Bienaym\'e-Galton-Watson tree, we give an exact characterisation of the critical value $\theta_c$ such that there is accessibility percolation with positive probability if and only if $\theta>\theta_c$. We also give more explicit bounds on the value of $\theta_c$. 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 $\theta_c$. To do this we introduce a novel coupling with oriented percolation.