Fixed points and cycle structure of random permutations

Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions. In particular this covers random permutations generated from Mallows Model with Kendall’s Tau, $\mu$ random permutations introduced in [11], as well as a class of exponential families introduced in [15].