By Rudenskaya O.G.
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Additional info for 4-Quasiperiodic Functions on Graphs and Hypergraphs
Let c > 0 be ﬁxed. For any η > 0 the bounds ρ(c) − η n ≤ C1 G(n, c/n) ≤ ρ(c) + η n hold whp as n → ∞. In other words, the normalized size of the giant component of G(n, c/n) converges in probability to ρ(c). g. that if c < 1 is constant then there is an A = A(c) such that C1 G(n, c/n) ≤ A log n holds whp. Proof. Let Gn = G(n, c/n). For each ﬁxed k, from Lemma 2 we have p 1 n Nk (Gn ) → P ( X(c) = k ). Deﬁning N<ω (Gn ) and N≥ω (Gn ) in the obvious way, it follows that there is some function ω = ω(n) tending to 45 Random Graphs and Branching Processes inﬁnity, which we may take to be o(n), so that 1 1 N≥ω (Gn ) − P ( X(c) ≥ ω ) = N<ω (Gn ) − P ( X(c) < ω ) n n ω−1 = k=1 1 p Nk (Gn ) − P ( X(c) = k ) → 0.
Set L = ε3 n, as before. Proof of Theorem 7. Let (31) A1 = log L A0 = log L − (5/2) log log L − K, and where K = K(n) → ∞ but K ≤ (1/3) log log L, say, so A0 → ∞. Note that (32) −5/2 −A0 LA0 e ∼ eK → ∞, while (33) −5/2 −A1 LA1 e = (log L)−5/2 → 0. For i = 0, 1, let ki = Ai /δ, where δ = δ(1 − ε) is deﬁned by (21), and let S+ = kTk k0 ≤k≤k1 58 B. Bollob´ as and O. Riordan be the number of vertices of Gn = G(n, (1 − ε)/n) in tree components of order between k0 and k1 . The required lower bound follows if we can show that S+ ≥ 1 whp.
For example, Theorem 7c of  asserts that if np(n) ∼ 1 and ω(n) → ∞, then whp the largest tree component of G n, p(n) has at least n2/3 /ω(n) and at most ω(n)n2/3 vertices. ) In fact, in this range the order of the largest tree component depends very strongly on the ‘error term’ ε = ε(n) = p(n)n − 1. 4, setting p = p(n) = 1 + ε(n) /n, with ε = −1/(log n)2 , say, the order of the largest tree component in G(n, p) is about 2(log n)5 , if ε = −n−1/5 then it is about 4(log n)n2/5 /5, and if ε = +n−1/5 then about 2n4/5 .