Random planar graphs

We study various properties of the random planar graph R n , drawn uniformly at random from the class P n of all simple planar graphs on n labelled vertices. In particular, we show that the probability that R n is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability R n has linearly many vertices of a given degree, in each embedding R n has linearly many faces of a given size, and R n has exponentially many automorphisms.