The Cycle Structure of Random Permutations
Richard Arratia,Simon Tavaré +1 more
TLDR
In this paper, the authors give a simple upper bound on the total variation distance between a random permutation and an independent Poisson process and show that this distance decays to zero superexponentially fast as a function of n/b \rightarrow \infty.Abstract:
The total variation distance between the process which counts cycles of size $1,2,\ldots, b$ of a random permutation of $n$ objects and a process $(Z_1,Z_2,\ldots, Z_b)$ of independent Poisson random variables with $\mathbb{E}Z_i = 1/i$ converges to 0 if and only if $b/n \rightarrow 0$. This Poisson approximation can be used to give simple proofs of limit theorems and bounds for a wide variety of functionals of random permutations. These limit theorems include the Erdos-Turan theorem for the asymptotic normality of the log of the order of a random permutation, and the DeLaurentis-Pittel functional central limit theorem for the cycle sizes. We give a simple explicit upper bound on the total variation distance to show that this distance decays to zero superexponentially fast as a function of $n/b \rightarrow \infty$. A similar result holds for derangements and, more generally, for permutations conditioned to have given numbers of cycles of various sizes. Comparison results are included to show that in approximating the cycle structure by an independent Poisson process the main discrepancy arises from independence rather than from Poisson marginals.read more
Citations
More filters
Book
Logarithmic Combinatorial Structures: A Probabilistic Approach
TL;DR: In this article, the authors explain the similarities in asymptotic behaviour as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition.
MonographDOI
Introduction to random graphs
Alan Frieze,Michał Karoński +1 more
TL;DR: All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
Journal ArticleDOI
On the Eigenvalues of Random Matrices
TL;DR: In this paper, it was shown that the set of eigenvalues of a random matrix chosen from Haar measure on the unitary group Un is independent and distributed as √jZ asymptotically as n → ∞.
Book ChapterDOI
Some developments of the Blackwell-MacQueen urn scheme
TL;DR: The Blackwell-MacQueen description of sampling from a Dirichlet random distribution on an abstract space is reviewed and extended to a general family of random discrete distributions in this paper, and results are obtained by application of Kingman's theory of partition structures.
Book
Asymptotic enumeration methods
TL;DR: In this article, the saddle point and large singularities of analytic functions are used for graphical enumeration, including implicit functions, recurrences, and combinations of methods, which are related to our work.