A lower bound for point-to-point connection probabilities in critical percolation
Consider critical site percolation on Zd with d≥2. We prove a lower bound of order n−d2 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, 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.