Researchers have calculated the exact 'tipping point' for a slope that allows a path to climb uphill forever in a random landscape.
April 1, 2026
Original Paper
Accessibility Percolation with Rough Mount Fuji labels
arXiv · 2603.29561
The Takeaway
In a world where ground height changes randomly at every step, it was unknown if a simple tilt was enough to guarantee you'd never get stuck at a dead-end peak. This proof defines the exact threshold where a landscape's incline ensures an infinite path upward, explaining how systems like evolving viruses or learning AIs can continue to improve indefinitely.
From the 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 $\theta$ 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 Bienaymé-Galton-Watson tree, we give an exact characte