# Random combinatorial structures: the convergent case

Barbour, A D; Granovsky, B (2005). Random combinatorial structures: the convergent case. Journal of Combinatorial Theory, Series A, 109(2):203-220.

## 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 independent random variables Zj, j ≥ 1, whose joint distribution, conditional on the event that Σnj=1 jZj = 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 Zj components of small sizes j, j ≥ 1, with the remaining part, of size close to n, being made up of a single, giant component.

## 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 independent random variables Zj, j ≥ 1, whose joint distribution, conditional on the event that Σnj=1 jZj = 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 Zj components of small sizes j, j ≥ 1, with the remaining part, of size close to n, being made up of a single, giant component.

## Statistics

### Citations

12 citations in Web of Science®
12 citations in Scopus®