Degree Sequence of Random Permutation Graphs

TL;DR

This paper investigates the degree sequences of permutation graphs generated by random permutations, establishing their limiting distributions, including uniform and Mallows permutations, with notable phase transitions and CLTs.

Contribution

It provides the first detailed analysis of degree distributions in permutation graphs for various random permutation models, including uniform and Mallows, with new limit theorems and phase transition insights.

Findings

Joint degree distributions converge to independent uniforms for uniform permutations.

Degree of the mid-vertex follows a central limit theorem.

Minimum degree converges to a Rayleigh distribution.

Abstract

In this paper we study the degree sequence of the permutation graph $G_{\pi_n}$ associated with a sequence $\pi_n\in S_n$ of random permutations. Joint limiting distributions of the degrees are established using results from graph and permutation limit theories. In particular, for the uniform random permutation, the joint distribution of the degrees of the vertices labelled $\lceil nr_1 \rceil, \lceil nr_2 \rceil, \ldots, \lceil nr_s \rceil$ converges (after scaling by $n$) to independent random variables $D_1, D_2, \ldots, D_s$, where $D_i\sim \text{Unif}(r_i, 1-r_i)$, for $r_i\in [0,1]$ and $i\in \{1, 2, \ldots, s\}$. Moreover, the degree of the mid-vertex (the vertex labelled $n/2$) has a central limit theorem, and the minimum degree converges to a Rayleigh distribution after appropriate scalings. Finally, the limiting degree distribution of the permutation graph associated with a Mallows random permutation is determined, and interesting phase transitions are observed. Our results extend to other exponential measures on permutations.