Degree Sequence of Random Permutation Graphs by Bhaswar B. Bhattacharya
نویسنده
چکیده
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 nr1 , nr2 , . . . , nrs in the associated permutation graph converges to independent random variables D(r1),D(r2), . . . ,D(rs), where D(ri)∼ Unif(ri ,1 − ri ), for ri ∈ [0,1] and i ∈ {1,2, . . . , 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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تاریخ انتشار 2017