Open Access
Translator Disclaimer
February 2017 Degree sequence of random permutation graphs
Bhaswar B. Bhattacharya, Sumit Mukherjee
Ann. Appl. Probab. 27(1): 439-484 (February 2017). DOI: 10.1214/16-AAP1207

Abstract

In this paper, we study the asymptotics of the degree sequence of permutation graphs associated with a sequence of random permutations. The limiting finite-dimensional distributions of the degree proportions are established using results from graph and permutation limit theories. In particular, we show that for a uniform random permutation, the joint distribution of the degree proportions of the vertices labeled $\lceil nr_{1}\rceil,\lceil nr_{2}\rceil,\ldots,\lceil nr_{s}\rceil$ in the associated permutation graph converges to independent random variables $D(r_{1}),D(r_{2}),\ldots,D(r_{s})$, where $D(r_{i})\sim\operatorname{Unif}(r_{i},1-r_{i})$, for $r_{i}\in[0,1]$ and $i\in\{1,2,\ldots,s\}$. Moreover, the degree proportion of the mid-vertex (the vertex labeled $n/2$) has a central limit theorem, and the minimum degree converges to a Rayleigh distribution after an appropriate scaling. Finally, the asymptotic finite-dimensional distributions of the permutation graph associated with a Mallows random permutation is determined, and interesting phase transitions are observed. Our results extend to other nonuniform measures on permutations as well.

Citation

Download Citation

Bhaswar B. Bhattacharya. Sumit Mukherjee. "Degree sequence of random permutation graphs." Ann. Appl. Probab. 27 (1) 439 - 484, February 2017. https://doi.org/10.1214/16-AAP1207

Information

Received: 1 April 2015; Revised: 1 March 2016; Published: February 2017
First available in Project Euclid: 6 March 2017

zbMATH: 1360.05150
MathSciNet: MR3619792
Digital Object Identifier: 10.1214/16-AAP1207

Subjects:
Primary: 05A05 , 60F05
Secondary: 60C05

Keywords: combinatorial probability , graph limit , limit theorems , Mallow’s model , permutation limit

Rights: Copyright © 2017 Institute of Mathematical Statistics

JOURNAL ARTICLE
46 PAGES


SHARE
Vol.27 • No. 1 • February 2017
Back to Top