Cycle length distributions in random permutations with diverging cycle weights

We study the model of random permutations with diverging cycle weights, which was recently considered by Ercolani and Ueltschi, and others. Assuming only regular variation of the cycle weights we obtain a very precise local limit theorem for the size of a typical cycle, and use this to show that the empirical distribution of properly rescaled cycle lengths converges in probability to a gamma distribution.Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,635–650, 2015


INTRODUCTION
We study the empirical cycle distributions in models of random permutations with weights depending on the length of the cycles. In this model, for any cycle of length j the weight of the permutation gets multiplied with a factor proportional to θ j . More precisely, the probability of a permutation π of n elements is defined as P n (π ) = 1 h n n! j≥1 θ r j (π) j where r j (π ) is the number of cycles in the permutation π of length j, and h n is a normalisation. The case of constant cycle weights θ j = θ corresponds to the Ewens measure from population biology and is well studied. In this paper our focus is on cycle weights (θ j ) which form a diverging sequence of regular variation. Studying random permutation with cycle weights described by their asymptotic behaviour was considered in [1] and is also motivated by the study of the quantum Bose gas [3,4]. The case of convergent sequences (θ j ) has also been studied, see e.g. [1,5,10].
The case of diverging cycle weights was treated by Betz et al. [5], Ercolani and Ueltschi [7], Nikeghbali and Zeindler [12] and by Maples et al. [11]. If the growth of the cycle weights is of polynomial order the length of a typical cycle goes to zero. Moreover, Ercolani and Ueltschi [7] show for a particular choice of the sequence (θ j ) that the length L 1 of the cycle containing one, behaves like where γ := lim log θ j log j > 0 and X is gamma distributed with shape parameter γ + 1. The aim of this paper is to generalise this result in several ways. First we allow for completely general sequences (θ j ) of regular variation with positive index, going well beyond the setting of [7]. See [2] for definitions and general results on this class of sequences. Second, we considerably refine the asymptotics and obtain a full local limit theorem. And third, building on this result, we extend the convergence to full convergence of the empirical cycle length distribution to a gamma distribution. The latter fact brings this result in line with similar results obtained in the study of condensation phenomena recently obtained by the authors in [6] and (S. Dereich, In preparation).
While the studies carried out for this model so far rely on the (often quite heavy) machinery of analytic combinatorics, like saddle-point analysis [7], singularity analysis [12] or further generating function methods [11], our proofs rely on a direct analysis of the renewaltype equations relating the normalisation factors h n to the cycle weights. The flexibility of this method is due to the fact that no inversion of generating functions has to be performed. One can expect that this method can also be used to extend further results from [7] and other papers in this area.

STATEMENT OF THE MAIN RESULTS
Recall the definition (1) of the random partitions and impose the following assumptions on the sequence (θ j ) of cycle weights: (A1) (θ j ) is regularly varying, i.e. there exists an index γ > 0 and a slowly varying function such that θ j = j γ (j) for all j ∈ N.
We let β 0 = 0 and β n := n j=1 θ j for integers n ≥ 1. By Lemma 3.4 below we have and, denoting by or in other words that (β ← (t)) is regularly varying with index 1 γ +1 . Finally, define and recall that (γ + 1) x γ e −dγ x , x ≥ 0 is the probability density of a gamma distribution with shape parameter γ + 1. We denote by L k = L k (σ ), the length of the cycle containing the symbol k ∈ {1, . . . , n}. The following local limit theorem is the first main result of this paper.
Theorem 2.1 (Local limit theorem). For every M > 0 we have Theorem 2.1 implies that a typical cycle under P n has length of order β ← (n). The following corollary is a slightly weaker version of Theorem 2.1, which is more illuminating in the case that j is of the order of a typical cycle length, and readily implies a global limit theorem.

Corollary 2.2. For every M > 0 we have
and therefore we have the global limit theorem where X is gamma distributed with shape parameter γ + 1.
We now define the empirical cycle length distribution as the random probability measure on [0, 1] given by where the integers λ 1 ≥ λ 2 ≥ · · · are the ordered cycle lengths of a permutation chosen randomly according to P n . We derive a limit theorem for the empirical cycle length distribution, showing that it converges in probability to a deterministic limit given by a gamma distribution. Our interest in Theorem 2.3 stems mostly from the analogy to results on the emergence of condensation, which also exhibit an incomplete gamma function describing the empirical distribution of a condensing quantity prior to condensation, see [6] for a speculative treatment of this universal phenomenon and results in the case of Kingman's model of selection and mutation and (S. Dereich. In preparation) for results on random networks.

Some First Observations
The following two lemmas hold without any assumptions on (θ j ). Crucial in the analysis of the model is the sequence (h n ) n≥0 of normalisations.
(b) The law of L 1 is given by Proof. See also Proposition 2.1 in [7]. We have that (a) follows from (b) by summing over all j ∈ {1, . . . , n}. For (b) we first sum over all the possible elements of the cycle containing one, in order, and then look at all the permutations of the remaining indices. This yields

Lemma 3.2.
Given the cycle containing one, the conditional distribution of the permutation on the remaining indices is given by P n−L 1 .
Proof. Note that the number of possible cycles of length l containing one is (n − 1)(n − 2) · · · (n − l + 1), and by Lemma 3.1 (b) the law of L 1 is given as Hence the conditional weight of any permutation σ containing the given cycle is whereσ is obtained from σ by removing the cycle containing one and relabelling the remaining indices as {1, . . . , n − l}.
The next lemma is a simple consequence of assumption (A2).
We collect relevant asymptotic estimates in the following lemma.

Proof of Theorem 2.1
The key to our analysis is to study the asymptotic behaviour of the normalising sequence (h n ) using the recurrence relation (2). Our main technical step, Proposition 3.9, shows that defining with the convention that h n = 0 for n ∈ −N, we have uniformly in the values a, b taken from a compact interval.
Let us first see how Theorem 2.1 follows from this. By Lemma 3.
The additional statement of Theorem 2.1 will be proved in Subsection 3.2.1. It constitutes the first step in the proof of (4), which will be carried out in four steps in Subsections 3.2.1 to 3.2.4. Subsection 3.2.5 is devoted to the proof of Corollary 2.2.

The Recurrence Equation.
In this section we show that for every This is a direct consequence of the following lemma. Indeed, using Lemma 3.1(b), Lemma 3.3, and Lemma 3.5 we get for every ε > 0 some M and n 0 such that, for all n ≥ n 0 , Proof. We analyse the sequence (h n ) at a large reference time N ∈ N. By Lemma 3.4, there exists n 0 ∈ N such that for any n ≥ n 0 and we define a sequence (α (N) k ) inductively by letting α (N) Random Structures and Algorithms DOI 10.1002/rsa for k = 0, . . . , K, and the remainder I (N) First we prove that, for k = 1, . . . , K, and β β ← (2n) ≥ 2n by definition, we get (conveniently dropping the round-off symbols in the summation) Next, let n = α (N) k + 1. By assumption n − 1 ≥ n 0 so that n − β ← (2n) ≥ α (N) k+1 and as above Similarly, it follows by induction over n, that where we used that α (N) k /α (N) k+1 ≤ 2 for k ∈ {0, . . . , K}, by construction. The estimate remains true for n ∈ I (N) 0 and, for n ∈ I (N) K+1 \{0}, one has Since β ← (2n 0 ) ≤ n 0 /2, one has β n 0 /2 = by (7). Altogether, we get that there is a constant c only depending on n 0 such that, for k ∈ {0, . . . , K + 1} and n ∈ I (N) k , Fix a constant M ∈ 2N and analyse which is the unique index l for which one has N − j ∈ I (N) l . By (8), one has Since, for k = 0, . . . , K + 1, Therefore, as long as β ← (2N) ≥ n 0 , one has Clearly, one has β(β ← (2N)) ∼ 2N as N → ∞. Further, the Potter bound [2, Theorem 1.5.6] implies that for sufficiently large n and any m ≥ n one has Consequently, one gets that, for sufficiently large N, The statement follows by choosing M sufficiently large.

Estimates Against the Volterra Equation.
Our aim is to show that g (N) , as defined in (3) Proof. We only prove the first statement, as the second can be proved analogously. Fix ε ∈ (0, 1/2) and choose M > 0 according to Lemma 3.5. In the following, we denote by 0 < ι 1 < ι 2 < . . . constants that can be chosen arbitrarily small and that do not depend on N and t. The following estimates are valid for sufficiently large N and all t ∈ [−L, L]. We let K ∈ N and set δ = M/K. Applying Lemma 3.5, we get For large N, one has N/(N + −Lβ ← (N) ) ≤ e ι 1 so that By definition of g (N) t , one has . Here we used that g (N) t−· is constant on intervals of length β ← (N). Hence, We note that, for each k = 1, . . . , K, Further, by Lemma 3.4, we have So far we have not imposed any assumptions on the positive constants ε and δ. We now assume that K is sufficiently large (or, equivalently, δ = M/K is sufficiently small) in order to guarantee existence of a nonnegative integer K 0 < K with One has Further, a K 0 → K 0 δ uniformly in t as N → ∞ and assuming that e ι 2 < 2, we conclude with the definition of K 0 , that, for N sufficiently large, Combining this with (9) and the estimate e ι 2 k γ /(k − 1) γ ≤ e ι 3 for k > K 0 , yields that where we used that which is a consequence of the uniform convergence a k → kδ as N → ∞.
Finally, we subtract 2εg (N) t , divide by 1 − 2ε to deduce that for all sufficiently large N and all t ∈ [−L, L]

Analysis of the Volterra Equation.
Lemma 3.6 relates our problem to the Volterra equation where ε ∈ R, T ∈ R, k ε (u) = e ε (1 + γ )u γ , for u ≥ 0, and f : [T , ∞) → R denotes a locally integrable function, see Remark 3.8 below for more details on this relation. We now collect some facts about this equation taken from [9,Chapter 2]. We only consider the case T = 0, since the general case can be easily obtained from the particular case by applying a time change. Further we write g = g 0 and k = k 0 . The unique solution to (10) can be expressed in terms of a fundamental solution. It is the unique solution to Again we abbreviate r = r 0 . With the fundamental solution we can represent the unique solution g ε to (10) as We will make use of the following properties.
(2) We have r(t) ∼ μ −1 e dγ t as t → ∞ where μ : Proof. (1) is easy to verify. For (2) we multiply (11) (with ε = 0) by e −dγ t and observe that the structure of the equation is retained with a new kernelk(u) := e −dγ u k(u), which is directly Riemann integrable and defines a probability density on the positive halfline. Hence, by the renewal theorem for densities (see for instance the 'alternative form' of the renewal theorem in [8, XI.1]), one has for the corresponding fundamental solutionr(t) = e −dγ t r(t) that lim t→∞r (t) = μ −1 , as required. As this is a Volterra equation we use the above representation of its solution to get that, for sufficiently large N, for t ∈ [−L, L]. Analogously, we obtain that, for sufficiently large N and t ∈ [−L, L],

Exponential behaviour of g (N)
t . In this section we finish the proof of (4) and hence of Theorem 2.1. We achieve this by combining the approximation and the results on the Volterra equation. Proposition 3.9. Let L, δ > 0. One has, for sufficiently large N ∈ N, that Proof. Given ε > 0 and κ ≥ 2L we define F ε as in Remark 3.8 and note that F (N,L,κ) We use the properties of the fundamental solution provided by Lemma 3.7 to rephrase (12) as follows We start with the derivation of an upper bound. Let δ ∈ (0, 1] be arbitrary. We will suppose that ε ∈ (0, δ] is a sufficiently small parameter, the actual value of which will be chosen later in the discussion. This choice may depend on L and κ but not on N or t. Assuming that ε ≤ δ we get that, for sufficiently large N, By Lemma 3.7, there exists T > 0 only depending on δ such that We restrict attention to t ∈ [−L + T , L]. We split the integral in (13) into two parts. The dominant part is Assuming that (e ε/(γ +1) − 1)2Ld γ ≤ δ we arrive at In order to show that the remaining part of the integral isasymptotically negligible, we first derive an estimate for F (N,L,κ) (s) for s ∈ [−L + 1, L]. One has u du and we observe that for the relevant values of u we have where we have used that s + L ≥ 1 and that the numerator is nonnegative. Hence, Consider now the remaining part of the integral in (13) where C T ≥ 1 is a constant only depending on T but not on the choice of L, κ, δ and ε. Combining (13) with (14) and (15)  An analogous lower bound can be proved similarly. By switching variables we get that, for δ ∈ (0, 1] arbitrary, there exist T , T > 0 such that for any L > 0 and sufficiently large κ ≥ 2L one has, for N sufficiently large and t ∈ [−L + T , L], This implies that, for −L + T ≤ a < b ≤ L, finishing the proof.

Proof of Corollary 2.2.
Using regular variation of (θ j ) and Lemma 3.4 (iii), for any κ > 0, Plugging this into Theorem 2.1 with M in the role of κ gives Hence we are done with the local result once we show that Note that if j/β ← (n) goes to zero, the second term inside the supremum vanishes asymptotically, and so does the first term by an application of (16) with an arbitrarily small value of κ > 0. Hence we can assume that the supremum is over εβ ← (n) ≤ j ≤ Mβ ← (n), for some fixed ε > 0. But on this domain we can exploit again that (θ j ) is regularly varying and Lemma 3.4 (iii) to obtain uniformly on the domain, which completes the proof of the local result in Corollary 2.2.
To infer that this implies the global limit theorem we observe that The term in brackets is a Riemann sum and therefore asymptotically equal to (γ + 1) x 0 y γ e −dγ y dy, which is the distribution function of a gamma distribution with shape parameter γ + 1.
Random Structures and Algorithms DOI 10.1002/rsa

Proof of Theorem 2.3
We now derive Theorem 2.3 from Corollary 2.2 using the first two moments of μ n [0, x], for fixed x > 0. The first moment is By Corollary 2.2 we have, To estimate the main term we use Lemma 3.2 to see that which implies that the variance of μ n [0, x] goes to zero. Hence the convergence in Theorem 2.3 holds in the L 2 sense, completing its proof.