On the connected components of a random permutation graph with a given number of edges

Abstract

A permutation of [n] induces a graph on [n] such that the edges of the graph correspond to inversion pairs of the permutation. This graph is connected if and only if the corresponding permutation is indecomposable. Let s(n,m) denote a permutation chosen uniformly at random among all permutations of [n] with exactly m inversions. Let p(n,m) be the common value for the probabilities that s(n,m) is indecomposable or the corresponding graph is connected. We prove that p(n,m) is non-decreasing with m by constructing a Markov process in which s(n,m+1) is obtained from s(n,m) by increasing one of the components of the inversion sequence of s(n,m) by one. We show that, with probability approaching 1, the graph corresponding to s(n,m) becomes connected for m asymptotic to (6/(\pi^2))nln(n). More precisely, for m=(6n/(\pi^2)) [ln(n)+ lnln(n)/2+ ln(12)- ln(\pi)- 12/(\pi^2)+x_n], where |x_n|=o(lnlnln(n)), the number of components of the random graph is shown to be asymptotically 1+Poisson(e^{-x_n}). When x_n goes to negative infinity, the sizes of the largest and the smallest components, scaled by n, are asymptotic to the lengths of the largest and the smallest subintervals in a partition of [0,1] by [e^{-x_n}] randomly, and independently, scattered points.