Limits of Random Trees. II

Abstract

Local convergence of bounded degree graphs was introduced by Benjamini and Schramm. This result was extended further by Lyons to bounded average degree graphs. In this paper we study the convergence of random tree sequences with given degree distributions. Denote by ${\mathcal{D}_n}$ the set of possible degree sequences of a labeled tree on n nodes. Let D _n be a random variable on ${\mathcal{D}_n}$ and T(D _n ) be a uniform random labeled tree with degree sequence D _n . We show that the sequence T(D _n ) converges in probability if and only if ${{\bf D}_n \rightarrow {\bf D} = ({\bf D}(i))^{\infty}_{i=1}}$, where ${{\bf D}(i) \sim {\bf D}(j), \mathbb{E}({\bf D}(1)) = 2}$ and D(1) is a random variable on ${\mathbb{N}^+}$.