Degree sequence of random permutation graphs

Abstract

In this paper we study the degree sequence of the permutation graph Gπ n associated with a sequence πn ∈ Sn 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 nr1 , nr2 , . . . , nrs converges (after scaling by n) to independent random variables D1, D2, . . . , Ds, where Di ∼ Unif(ri, 1 − ri), for ri ∈ [0, 1] and i ∈ {1, 2, . . . , 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.2010 Mathematics Subject Classification. 05A05, 05C17, 60C05, 60F05. √ 2 (Theorem 3.7). Moreover, we give sufficient conditions for verifying the convergence of the permutation process. These conditions can be easily verified for many non-uniform (exponential) measures on permutations. These conditions together with the recent work of Starr [40] can be used to explicitly determine the limiting distribution of the degree process for a Mallows random permutation, for all β ∈ R (Theorem 3.8). For each a ∈ (0, 1], the limiting density of d n ( na )/n has a interesting phase transition depending on the value of β: there exists a critical value β c (a) such that for β ∈ [0, β c (a)] the limiting density is a continuous function supported on [a, 1 − a]. However, for β > β c (a) the density breaks into two piecewise continuous parts. If β = 1/T denotes the inverse temperature, then this is the statistical physics phenomenon of replica symmetry breaking in the low temperature regime.The rest of the paper is organized as follows: Section 2 contains the basics of graph and permutation limit theories and their connections. Section 3 gives the summary of our main results.1 Throughout the paper, πn will be used interchangeably to denote both the permutation and the permutation process depending on the context. In particular, for a ∈ [n] πn(a) will denote the image of a under the permutation πn. On the other hand, for t ∈ [0, 1] πn(t) = πn( nt )/n will denote the permutation process evaluated at t. Similarly, dn will be used to denote both the degree of a vertex and the degree process.

Fixed points and cycle structure of random permutations

“…of (2.8) are the same, and so gets canceled. As an example, this happens for the choice l = 3, k 1 = 0, k 2 = 1, k 3 = 2, p(2, 1) = (1, 2), p(3, 1) = (3, 4, 5), p(3, 2) = (3,4,5).…”

“…of (2.9) are 0. Even if L(D) = L, it is possible that both terms are 0, which happens for example for the choice l = 3, k 1 = 0, k 2 = 1, k 3 = 2, p(2, 1) = (1, 2), p(3, 1) = (3,5,4), p(3, 2) = (3,4,5).…”

“…Any µ ∈ M is called a permuton (following [11]), and it is shown in [11,Theorem 1.6] that any µ ∈ M can indeed arise as a limit of a sequence of permutations in this manner. See [3,11] for a more detailed introduction to permutation limits. If {π n } n≥1 is a sequence of random permutations (not necessarily in the same probability space), the sequence is said to converge to a deterministic measure µ ∈ M in probability, if the sequence of measures ν πn converge weakly to the measure µ in probability.…”

“…For every n ≥ 1 let π n be a random permutation on S n with law P n . In [3, Def 6.2] the authors define a notion of equi-continuity of random permutations, which they show is implied by the condition lim δ→0 lim n→∞ sup p,q,r∈S(n,l):||p−r||∞≤nδ P n (π n (p) = q) P n (π n (r) = q) [3,Prop 6.2]). In particular for l = 1 condition (1.3) in spirit demands that the function P n (π n (p) = q) is equi-continuous in p. In this paper we will need an extra notion of equi-continuity which demands that the function P n (π n (p) = q) is jointly equi-continuous in p, q.…”

A study on random permutation graphs

“…Afterwards, we focus on the asymptotic distribution of the degree of a given node. In particular, we provide a very simple proof for a central limit theorem for the mid-node which was previously proven in [8] by using different techniques. We do not restrict ourselves here just to the mid-node, but we also prove a central limit theorem for any given fixed node k as the number of nodes n grows -actually k is allowed to grow as well, see below for the exact statement.…”