Consider critical site percolation on Zd with d ≥ 2. We prove a lower bound of order n-d 2 for point-to-point connection probabilities, where n is the distance between the points. Most of the work in our proof concerns a ‘construction’ which finally reduces the problem to a topological one. This is then solved by applying a topological fact, Lemma 2.12 below, which follows from Brouwer’s fixed point theorem. Our bound improves the lower bound with exponent 2d(d-1), used by Cerf in 2015  to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in itself, our result gives a small improvement of the bound on the two-arm exponent found by Cerf.
- Connection probabilities
- Critical percolation