Existence of infinite accessible paths near the diagonal at θ=1 for ℓ^q metrics on ℤ2

Determine whether, in the Rough Mount Fuji accessibility percolation model on ℤ2 with vertex labels X_v = U_v + θ‖v‖_q, with q>1 and θ=1, there exists an infinite non-backtracking path from the origin along which the labels are strictly increasing that stays within a cone around the diagonal line y = x.

Background

The paper studies Rough Mount Fuji (RMF) accessibility percolation on ℤn, focusing on non-backtracking paths and distances given by ℓq norms. For q>1, neighboring vertices can have very small differences in their distances to the origin away from the axes, complicating couplings used for the ℓ1 case.

To prove percolation for θ close to 1 with q>1, the authors restrict attention to paths confined to a narrow cone near an axis and introduce a two-scale “bricklayer” coupling. They note, however, that although axes yield deterministically increasing paths when θ=1, it is not clear whether any infinite increasing path exists that stays near the diagonal, motivating this open question.

References

However, from a theoretical standpoint, it is not even clear that there are infinite accessible paths near the diagonal when θ=1, whereas along each axis, we clearly do have at least one accessible path when θ=1, and it is this property that draws us, in the proof, to restrict our class of paths to those remaining within a cone near the horizontal axis.

— Accessibility Percolation with Rough Mount Fuji labels  (2603.29561 - Bellon et al., 31 Mar 2026) in Results on ℤ^n (Introduction), paragraph preceding the three-step proof outline and Proposition \ref{prop:lattice_perc_l2}