Crossing probabilities for critical Bernoulli percolation on slabs

TL;DR

This paper proves that in critical Bernoulli percolation on lattice slabs, the probability of crossing rectangles horizontally remains uniformly positive regardless of the rectangle's aspect ratio.

Contribution

It establishes a uniform positivity result for crossing probabilities in critical percolation on lattice slabs, extending known planar results to slab geometries.

Findings

Open crossing probabilities are uniformly positive for all aspect ratios.

Results extend planar percolation theory to three-dimensional slab geometries.

Supports the universality of critical phenomena in percolation models.

Abstract

We prove that in the critical Bernoulli percolation on two dimensional lattice slabs the probabilities of open left-right crossings of rectangles with any given aspect ratio are uniformly positive.