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).
|Number of pages||18|
|Journal||Random Structures & Algorithms|
|Publication status||Published - Aug 1996|