We study site percolation on the hypercubic lattice, in which each site of the lattice is independently occupied with probability p, or vacant with probability 1 - p.
A pattern is a prescribed configuration of occupied and vacant sites in a d-dimensional cube of fixed diameter. We show that with high probability, every pattern appears with positive density on a big occupied percolation cluster. Moreover, with high probability, two distinct patterns must occur on a big cluster in a given ratio (which we identify explicitly).
This result leads to a new and simple proof of the ratio limit theorem for percolation, which states that the ratio of the probabilities that the occupied cluster of the origin has size n + 1 and size n, respectively, converges as n tends to infinity. A somewhat stronger result is obtained in the supercritical case.