Uniqueness of the Infinite Cluster and Related Results in Percolation

We present the following results for independent percolation: a)continuous differentiability (in the natural parameters) of the free energy function (mean number of clusters per site),b)uniqueness of the infinite cluster,c)continuity of the connectivity functions. As a corollary of a) and previous results of van den Berg and Keane, there followsd)continuity of the percolation density, except possibly at the critical point. These results are valid for both site and bond percolation on d-dimensional lattices with arbitrary d and for translation-invariant long range bond models (satisfying a natural irreducibility condition).