The perturbation method and triangle-free random graphs

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Abstract

When a discrete random variable in a discrete space is asymptotically Poisson, there is often a powerful method of estimating its distribution, by calculating the ratio of the probabilities of adjacent values of the variable. The versatility of this method is demonstrated by finding asymptotically the probability that a random graph has no triangles, provided the edge density is not too large. In particular, the probability that G ∈ script G sign(n, p) has no triangles is asymptotic to exp(-1/6 p3n3 + 1/4 p5n4 - 7/12 p7n5) for p = o(n-2/3), and for G ∈ script G sign(n, m) it is asymptotic to exp(-1/6 d3 n3) for d = 2m/n(n -1) = o(n-2/3).

Original languageEnglish
Pages (from-to)253-270
Number of pages18
JournalRandom Structures & Algorithms
Volume9
Issue number1
Publication statusPublished - Aug 1996
Externally publishedYes

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