The Second Moment Method, Conditioning and Approximation

The second moment method has been fruitfully combined with conditioning by several authors, for example Robinson and Wormald 7], 8] and Frieze and Janson 2]. We try to give a general exposition of this method, and of its relation to approximations of random variables by simpler variables. We discuss several examples where the method has been used in diierent forms. 1. The second moment method One of the classical methods to show the existence (with large probability) of certain substructures in a random graph is the second moment method. We can describe a general version of the method as follows. Consider some random combinatorial structure G. In order to study the existence of a certain type of substructure (for example, a Hamilton cycle, a perfect matching, an induced C 4 ; : : :), we let X = X(G) be the number of such substructures in G. Thus X is a random variable taking non-negative integer values, and the problem is to estimate P(X > 0). We assume that P(X > 0) > 0 (otherwise the substructures never appear) and that E X 2 < 1 (in fact, we usually study nite random structures, where X is bounded). Typically, we are interested in asymptotic results. We consider a sequence of such random structures G (a more precise notation would be G n , but we will usually ignore the subscript n) and want to show P(X > 0) ! 1.