Asymptotics for Lipschitz percolation above tilted planes

TL;DR

This paper investigates the asymptotic behavior of the critical probability in Lipschitz percolation above tilted planes in high dimensions and as the tilt angle approaches a quarter turn, revealing polynomial convergence rates.

Contribution

It provides the first detailed asymptotic analysis of the critical probability for Lipschitz percolation above tilted planes in high dimensions and for large tilt angles.

Findings

Critical probability converges polynomially to 1 as dimension increases.

Identifies the exact order of polynomial convergence.

Determines the precise prefactor in the one-dimensional case.

Abstract

We consider Lipschitz percolation in $d+1$ dimensions above planes tilted by an angle $\gamma$ along one or several coordinate axes. In particular, we are interested in the asymptotics of the critical probability as $d \to \infty$ as well as $\gamma \to \pi/4.$ Our principal results show that the convergence of the critical probability to 1 is polynomial as $d\to \infty$ and $\gamma \to \pi/4.$ In addition, we identify the correct order of this polynomial convergence and in $d=1$ we also obtain the correct prefactor.