Emergence of a giant component in a random permutation with given number of cycles

Kazimirov, N. I.

doi:10.1163/156939203322694781

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“…See further Kolchin, Sevast'yanov and Chistyakov [77] and Kolchin [76]. By Remark 10.8, we obtain results on random permutations with m cycles as m/n → λ ∈ [1, ∞), see for example Kazimirov [71]. However, it is of greater interest to consider random permutations without constraining the number of cycles.…”

Section: Examples Of Balls-in-boxesmentioning

confidence: 70%

Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation

Janson¹

2012

Probab. Surveys

193

367

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Simply generated trees and Galton-Watson treesWe suppose that we are given a fixed weight sequence w = (w k ) k 0 of non-negative real numbers. We then define the weight of a finite rooted and ordered (a.k.a. plane) tree T bytaking the product over all nodes v in T , where d + (v) is the outdegree of v. Trees with such weights are called simply generated trees and were introduced by Meir and Moon [24]. We let T n be the random simply generated tree obtained by picking a tree with n nodes at random with probability proportional to its weight. (To avoid trivialities, we assume that w 0 > 0 and that there exists some k 2 with w k > 0. We consider only n such that there exists some tree with n vertices and positive weight.)One particularly important case is when ∞ k=0 w k = 1, so the weight sequence (w k ) is a probability distribution on Z 0 . (We then say that (w k ) is a probability weight sequence.) In this case we let ξ be a random variable with the corresponding distribution: P(ξ = k) = w k . It is easily seen that the simply generated random tree T n equals the conditioned Galton-Watson tree with offspring distribution ξ, i.e., the random Galton-Watson tree defined by ξ conditioned on having exactly n vertices.One of the reasons for the interest in these trees is that many kinds of random trees occuring in various applications (random ordered trees, unordered trees, binary trees, . . . ) can be seen as simply generated random trees and conditioned Galton-Watson trees, see e.g. Aldous [3,4], Devroye [9] and Drmota [10].It is easily seen that if a, b > 0 and we change w k to

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Section: Examples Of Balls-in-boxesmentioning

confidence: 70%