The Annals of Applied Probability

Degree sequence of random permutation graphs

Bhaswar B. Bhattacharya and Sumit Mukherjee

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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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 1 (2017), 439-484.

Dates
Received: April 2015
Revised: March 2016
First available in Project Euclid: 6 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1488790832

Digital Object Identifier
doi:10.1214/16-AAP1207

Mathematical Reviews number (MathSciNet)
MR3619792

Zentralblatt MATH identifier
1360.05150

Subjects
Primary: 05A05: Permutations, words, matrices 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Keywords
Combinatorial probability graph limit limit theorems Mallow’s model permutation limit

Citation

Bhattacharya, Bhaswar B.; Mukherjee, Sumit. Degree sequence of random permutation graphs. Ann. Appl. Probab. 27 (2017), no. 1, 439--484. doi:10.1214/16-AAP1207. https://projecteuclid.org/euclid.aoap/1488790832


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