Fixed points and cycle structure of random permutations

TL;DR

This paper investigates the asymptotic distribution of fixed points and cycle structures in convergent sequences of random permutations, including models like Mallows and exponential families, using permutation limits.

Contribution

It introduces a unified approach to analyze the limiting cycle structure of various random permutation models via permutation limits.

Findings

Derived the limiting distribution of fixed points for convergent permutation sequences.

Established the cycle structure distribution for models like Mallows and exponential families.

Unified analysis framework applicable to multiple permutation models.

Abstract

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].