Random planar graphs with n nodes and a fixed number of edges

Let <i>P</i>(<i>n, m</i>) be the class of simple labelled planar graphs with <i>n</i> nodes and <i>m</i> edges, and let <i>R</i><inf>n,q</inf> be a graph drawn uniformly at random from <i>P</i>(<i>n</i>, [<i>qn</i>]). We show properties that hold with high probability (w.h.p.) for <i>R<inf>n,q</inf></i> when 1 < <i>q</i> < 3. For example, we show that <i>R<inf>n,q</inf></i> contains w.h.p. linearly many nodes of each given degree and linearly many node disjoint copies of each given fixed connected planar graph. Additionally, we show that the probability that <i>R<inf>n,q</inf></i> is connected is bounded away from one by a non-zero constant. As a tool we show that (|<i>P</i>(<i>n</i>, [<i>qn</i>])|/<i>n!</i>)^1/n tends to a limit as <i>n</i> tends to infinity.