Growth of the number of spanning trees of the Erdo″s-Rényi giant component

Russell Lyons*, Ron Peled, Oded Schramm

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The number of spanning trees in the giant component of the random graph g (n, c/n) (c > 1) grows like exp{m(f(c)+o(1))} as n → ∞, where m is the number of vertices in the giant component. The function f is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on f'(c). A key lemma is the following. Let PGW(δ) denote a Galton-Watson tree having Poisson offspring distribution with parameter δ. Suppose that δ * >δ>1. We show that PGW(δ*) conditioned to survive forever stochastically dominates PGW(δ) conditioned to survive forever.

Original languageEnglish
Pages (from-to)711-726
Number of pages16
JournalCombinatorics Probability and Computing
Volume17
Issue number5
DOIs
StatePublished - Sep 2008
Externally publishedYes

Funding

FundersFunder number
Directorate for Mathematical and Physical Sciences0705518, 0406017, 0605166

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