GAUSSIAN FLUCTUATIONS OF CONNECTIVITIES IN THE SUBCRITICAL REGIME OF PERCOLATION

We consider the d-dimensional Bernoulli bond percolation model and prove the following results for all p < p(c): (1) The leading power-law correction to exponential decay of the connectivity function between the origin and the point (L, 0, ..., 0) is L-(d-1)/2. (2) The correlation length, zeta (p), is real analytic. (3) Conditioned on the existence of a path between the origin and the point (L, 0, ..., 0), the hitting distribution of the cluster in the intermediate planes, x1 = qL, 0 < q < 1, obeys a multidimensional local limit theorem. Furthermore, for the two-dimensional percolation system, we prove the absence of a roughening transition: For all p > p(c), the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge - in the infinite-volume limit - to the standard Bernoulli measure.