TY - JOUR

T1 - Growth of the number of spanning trees of the Erdo″s-Rényi giant component

AU - Lyons, Russell

AU - Peled, Ron

AU - Schramm, Oded

PY - 2008/9

Y1 - 2008/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=51349165867&partnerID=8YFLogxK

U2 - 10.1017/S0963548308009188

DO - 10.1017/S0963548308009188

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AN - SCOPUS:51349165867

SN - 0963-5483

VL - 17

SP - 711

EP - 726

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 5

ER -