Random Graphs

Once upon a time, I began as an analyst. I studied function spaces such as BMO and operators on function spaces, see for example 6,7], but gradually I became more and more interested in probability theory. Some years ago I worked on random coverings and other problems in geometrical probability 8,9], and at present most of my time is devoted to the related eld of combinatorial probability, in particular random graphs. This may seem to be very diierent from the harmonic and functional analysis that I once worked on (and still continue with sometimes), but the diierence in methods is not so great. I study random graphs as a probabilist dealing with some combinatorial structures, and my methods are probabilistic and based on analysis, using for example integration theory, functional analysis, martingales and stochastic integration. In this presentation I will give a survey over some recent results on random graphs where I have been at least partly involved. The systematic study of random graphs was started by Erd} os and Rnyi 4] in 1960, and the theory has expanded rapidly during the last decade. For a fuller historical account, and for many other results on random graphs, I refer to Bollobs's book 3]. Definitions A random graph is a graph generated by some random procedure. There are many (non-equivalent) ways to deene random graphs. The simplest, denoted by G n;m (or one of several common similar notations), where n and m are two integers with 0 m ? n 2 , is obtained by taking a set of n elements as the set of vertices, for deeniteness we may take the integers 1; : : : ; n, and then randomly selecting (by drawing without replacement) m of the ? n 2 possible edges. A closely related model, denoted by for example G n;p , where 0 p 1, is obtained by taking the same vertex set but now selecting every possible edge with probability p, independently of all other edges. (In particular, p = 1=2 gives the uniform distribution over all (labelled) graphs on n vertices.) We are mainly interested in the case when n, the number of vertices, is very large, and especially in asymptotic results when n ! 1 and m or p is a given function of n. Note that there is a priori no structure in these models. All vertices and pairs of vertices are …