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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


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.


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Bhaswar B. Bhattacharya. Sumit Mukherjee. "Degree sequence of random permutation graphs." Ann. Appl. Probab. 27 (1) 439 - 484, February 2017.


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

Primary: 05A05 , 60F05
Secondary: 60C05

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

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 1 • February 2017
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