Gaussian fluctuations of connectivities in the subcritical regime of percolation (Q582721)

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, \(\xi\) (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, \(x_ 1=qL\), \(0<q<1\), 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.