An exact universal amplitude ratio for percolation

The universal amplitude ratio R̃ ξ for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model, with q 6 4. The numerical result obtained by Delfino and Cardy by quantum field theory techniques agrees well with the algebraic expression determined here. Percolation is a subject which can be described simply and yet has many important physical applications (see [1] and references therein). In statistical mechanics, one natural arena in which to study its inherent critical phenomena, percolation is related by way of the random cluster model [2] to the q-state Potts model with q = 1. The critical exponents of the two-dimensional Potts model for q 6 4, determined from numerical and renomalization group studies [3, 4], led to the identification [5] of the Potts model (in the scaling limit) for q = 2, 3, 4 with the φ2,1 perturbation of the minimal unitary series M(t, t+ 1) by way of √ q = 2 sin ( π(t− 1) 2(t+ 1) )