Spanning Trees, Random Graphs, and Random Walks

In the usual Erdõs-Rényi model of random graphs, each pair of n vertices is connected by an edge independently with probability c/n for some constant c. When c > 1, it has a unique 'giant' component. How quickly does the number of spanning trees of the giant component grow with n compared to the growth in the number of its vertices? Is it monotonic in c? We answer this in joint work with Ron Peled and Oded Schramm.