Compatible sequences and a slow Winkler percolation

TL;DR

This paper investigates the compatibility of two infinite random 0-1 sequences, demonstrating that for small enough probability p of 1's, they are compatible with positive probability, revealing a novel percolation behavior.

Contribution

It answers a question by Peter Winkler by showing that random i.i.d. sequences with small p are compatible with positive probability, highlighting a unique polynomial decay in percolation.

Findings

Compatibility occurs with positive probability for small p

Percolation exhibits power-law decay, not exponential

Origin blocking probability decreases polynomially with distance

Abstract

Two infinite 0-1 sequences are called compatible when it is possible to cast out 0's from both in such a way that they become complementary to each other. Answering a question of Peter Winkler, we show that if the two 0-1-sequences are random i.i.d. and independent from each other, with probability p of 1's, then if p is sufficiently small they are compatible with positive probability. The question is equivalent to a certain dependent percolation with a power-law behavior: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially.