Random Combinatorial structures:the convergent case

Abstract

This paper studies the distribution of the component spectrum of combinatorial structures such as uniform random forests, in which the classical generating function for the numbers of (irreducible) elements of the different sizes converges at the radius of convergence; here, this property is expressed in terms of the expectations of {\it independent} random variables $Z_j$, $j\ge1$, whose joint distribution, conditional on the event that $\sum_{j=1}^n jZ_j = n$, gives the distribution of the component spectrum for a random structure of size $n$. For a large class of such structures, we show that the component spectrum is asymptotically composed of $Z_j$ components of size $j$, $j\ge1$, with the remaining part, of size $n-\sum_{j\ge1} Z_j$, being made up of a single, giant component.