On smooth mesoscopic linear statistics of the eigenvalues of random permutation matrices

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function $f$ to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $f(0)\neq 0$, we have a central limit theorem with a logarithmic variance, and if $f(0) = 0$, the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.


Introduction
The spectrum of random permutation matrices has been studied with much attention in the last few decades. On the one hand, working with matrices gives a different way to understand some of the classical properties satisfied by random permutations. On the other hand, the set of permutation matrices can be seen as a finite subgroup of the orthogonal group or the unitary group, and thus an interesting problem consists in studying how similar are the spectral behaviors of random permutations and usual ensembles of random orthogonal or unitary matrices. For a random permutation matrix following one of the Ewens measures, the number of eigenvalues lying on a fixed arc of the unit circle has been studied in detail by Wieand [34], and satisfies a central limit theorem when the order n goes to infinity, with a variance growing like log n. This rate of growth is similar to what is obtained for the Circular Unitary Ensemble and random matrices on other compact groups, for which a central limit theorem also occurs, as it can be seen in Costin and Lebowitz [11], Soshnikov [30] and Wieand [33]. A similar result has recently been proven by Bahier [6], on the number of eigenvalues lying on a mesoscopic arc, for a suitable modification of Ewens distributed permutation matrices, and the growth of the variance is also the same as for the CUE, i.e. the logarithm of n times the length of the interval. Some other results on the distribution of eigenvalues of matrices constructed from random permutations can be found in papers by Bahier [5], Evans [16], Najnudel and Nikeghbali [28], Tsou [31], Wieand [35].
The analogy between the permutation matrices and the CUE is not as strong when we consider smooth linear statistics of the eigenvalues. In this case, if we take a fixed, sufficiently smooth test function, it is known that the fluctuations of the corresponding linear statistics tend to a limiting distribution, without normalization, which is unusual for a limit theorem. In the CUE case, the distribution is Gaussian, as seen in Diaconis and Shahshahani [14], Johansson [22], Diaconis and Evans [13], and the variance is proportional to the squared H 1/2 norm of the test function. In the case of permutation matrices, the limiting distribution is not Gaussian anymore: its shape depends on the test function f and can be explicitly described in terms of f and a sequence of independent Poisson random variables. More detail can be found in Manstavicius [27], Ben Arous and Dang [7].
In the case of mesoscopic linear statistics, one also has a central limit theorem without normalization in the CUE case (see [29]). The behavior of mesoscopic linear statistics of other random matrix ensembles have also been studied: the Gaussian Unitary Ensemble (see [15]), more general Wigner matrices (see [20], [19]) and determinantal processes (see [23]), the Circular Beta Ensemble (see [25]), the thinned CUE, for which a random subset of the eigenvalues has been removed (see [8]). However, the smooth mesoscopic linear statistics of permutation matrices have not been previously studied. The main point of the present article is to show that they also satisfy some limit theorems.
The precise framework is given as follows. We fix a parameter θ > 0, and we consider a sequence (σ n ) n≥1 , σ n following the the Ewens(θ) distribution on the symmetric group S n , that is to say where K(σ) denotes the total number of cycles of σ once decomposed as a product of cycles with disjoint supports. Note that the particular case θ = 1 corresponds to the uniform distribution on S n . The permutation matrix M σ associated with any element σ of S n is defined as follows: for all 1 ≤ i, j ≤ n, A key relationship between the cycle structure of σ and the spectrum of the corresponding permutation matrix M σ appears in the expression of the characteristic polynomial of M σ : where a σ j denotes the number of j-cycles in the decomposition of σ as a product of disjoint cycles. As a consequence, the cycle structure of σ is fully determined by the spectrum of M σ , counted with multiplicity.
In this paper we are interested in the mesoscopic behavior of smooth linear statistics of the spectrum of M σn when n goes to infinity. More precisely, we fix a function f from R to C which satisfies the following regularity conditions: Moreover, we fix a sequence (δ n ) n≥1 in R * + such that δ n → 0 and nδ n → ∞ when n → ∞, which means that the corresponding scale is mesoscopic (small but large with respect to the average spacing between the eigenvalues of M σn ). In this article, we mainly study the following quantity: where S(σ n ) denotes the spectrum of M σn and m n (e ix ) is the multiplicity of e ix as an eigenvalue of M σn . In other words, we sum the function f at the eigenangles of M σn , divided by 2πδ n and counted with multiplicity. Notice that all the determinations of the eigenangles are considered here, and the set of x involved in the sum is 2π-periodic. Notice that the sum giving X σn,δn (f ) is absolutely convergent, because of the assumption we make on the decay of f at infinity. We will also consider the version of the linear statistics where we restrict the sum to the determinations of the eigenangles which are in the interval (−π, π]: In order to state our main theorem, we need to introduce the Fourier transform of f , normalized as follows: and the two following functions from R * + to C: and The series defining Θ f is absolutely convergent because of the assumptions (1). Our main result can now be stated as follows: Theorem 1.1. Let (δ n ) n≥1 be a positive sequence such that δ n −→ n→∞ 0 and nδ n −→ n→∞ ∞, and let f be a function from R to C satisfying the assumptions (1) given above.
(i) If f (0) = 0, then we have the following asymptotics: and Moreover, the following central limit theorem holds: (ii) If f (0) = 0, then we have the following convergence in distribution: where X is a Poisson point process with intensity θ x dx on (0, +∞), and where the sum on X in the right-hand side is a.s. absolutely convergent.
(iii) For any α > 1 such that (1) is satisfied, the results given in (i) and (ii) are still true if we replace X σn,δn (f ) by X ′ σn,δn (f ), as soon as In the spectrum of M σn , each cycle of length ℓ gives eigenangles equal to all multiples of 2π/ℓ. The contribution of these eigenangles in the sum X σn,δn (f ) is: Then, if we denote by a n,j the number of j-cycles in the decomposition of σ n as a product of cycles with disjoint support, we get: ℓδ n a n,ℓ .
Since the total number of elements of all cycles is n, we have n ℓ=1 ℓa n,ℓ = n, and then Note thatf (0) is the integral of f , and then the term nδ nf (0) is what we would obtain with a constant density of eigenangles of n/2π. (1), the Poisson summation formula applies and gives for all x > 0,

First note that under
We now get the following asymptotic result on Θ f : (i) Θ f is continuous on R * + and converges at infinity to f (0) with rate dominated by 1 x α (where α is given by (1)).
(ii) Θf is continuous on R * + and converges at infinity tof (0) with rate dominated by 1 x 2 . Proof. We prove the two items separately.
• Proof of (i): Since f is assumed to be continuous, the functions f k : x k f k converges uniformly on compact sets of (0, +∞). We deduce the continuity of Θ f . For the convergence of Θ f to f (0) at infinity, we only have to notice that for all x ≥ 1, • Proof of (ii): It is clear that the functions g k : x →f (kx) are continuous on (0, +∞) for all k ∈ Z, and from two consecutive integrations by parts, it follows that for all k ∈ Z \ {0} and for all x in any compact hence k g k converges uniformly on compact sets of (0, +∞). Note that there is no boundary term in the integration by parts, since by assumption, f goes to zero at infinity, and f ′ and f ′′ are integrable, which implies that f ′ also goes to zero at infinity.
and the proof is complete.
From the proposition just above, we deduce the following lemma: If the function f satisfies the assumptions (1), then for all In particular, Moreover, Ξ f (x) is continuous at any point of R * + \{1}, and also at 1 if f (0) = 0.
Proof. We have, for all x ∈ (0, 1], and for all x ∈ (1, ∞), The continuity of Ξ f is an immediate consequence of the continuity of Θ f .

The Feller coupling
In [18], Feller introduces a construction of a uniform permutation on the symmetric group, such that the cycle lengths are given by the spacings between successes in independent Bernoulli trials. This construction can be extended to general Ewens distributions, and provides a coupling between the cycle counts of a random permutation and a sequence of independent Poisson random variables. For the detail of the coupling procedure and many related results, we refer to [2] and [7,Section 4]. From the Feller coupling, we can deduce the following lemma: For all n, one can couple the numbers a n,ℓ of ℓ-cycles in the random permutation σ n , with a sequence of independent Poisson variables W ℓ of parameters θ/ℓ, in such a way that where C(θ) is a constant which does not depend on n.
It is then enough to bound the L 2 norm of G n − H n by a quantity depending only on θ, for G n := n j=1 a n,j , H n := n j=1 W j . Such a bound is a consequence of [7,Lemma 4.8], in the case where u j = 1 for 1 ≤ j ≤ n.
The lemma proven here allows to compare the quantity X σn,δn (f ) with a linear combination of independent Poisson random variables, for which classical tools in probability theory can be used to prove limit theorems.

Proof of Theorem 1.1 (i)
We couple the variables (a n,ℓ ) 1≤ℓ≤n with independent Poisson variables (W ℓ ) ℓ≥1 by using the Feller coupling, as in the previous section. From (2), we get (4) In order to prove the first part of Theorem 1.1, we will show that the sum of the two first terms satisfies the same central limit theorem, and that the two last term are bounded in L 2 . We first prove the following result: Proof. Since (W ℓ ) ℓ≥1 are independent Poisson random variables, W ℓ with parameter θ/ℓ, we get which gives the estimates of the proposition for the expectation and the variance. The central limit theorem is easily obtained by applying the Lindeberg-Feller criterion, since the variables (W ℓ ) ℓ≥1 are independent.
We then prove that the two last terms of (4) are bounded in L 2 : Proposition 4.2. We have the estimate: Moreover, we have a n,ℓ ≤ W ℓ for all ℓ except at most one value, for which we may have a n,ℓ = W ℓ + 1. It is then enough to check and These estimates are implied by the estimate which is a direct consequence of Lemma 2.2.
It is now easy to deduce Theorem 1.1 (i) from the two propositions just above. The estimate of the expectation is immediate, and the estimate of the variance is directly deduced from the fact that

Proof of Theorem 1.1 (ii)
Let A n := n ℓ=1 a n,ℓ Ξ f Here, a n,ℓ and W ℓ are again related by the Feller coupling. Notice that the sum defining Z is a.s. absolutely convergent, since We are going to prove the result in two steps: (ii) For all t ∈ R, E(e itAn ) − E(e itBn ) −→ n→∞ 0.
(i): Let t ∈ R. Using that the variables W j are independent, for any R > 1. For fixed R and n large enough depending on R, the condition 1 ≤ ℓ ≤ n can be discarded in the first exponential of the last product, since δ n → 0 and nδ n → ∞ when n → ∞. The sum in the first exponential is then a Riemann sum, which by continuity of Ξ f (proven in Lemma 2.2: recall that f (0) = 0 in this section), shows that the exponential tends to when n goes to infinity. On the other hand, by Lemma 2.2 the sum inside the second exponential is dominated by and then the second exponential is Since the left-hand side of the convergence is 1 + O f,θ,t (1/R), we deduce that Letting R → ∞, we deduce where the convergence of the integrals is insured by the integrability of |Ξ f (x)|dx/x given in Lemma 2.2. By Campbell's theorem, i.e. E(e itZ ) is the unique possible limit of a subsequence of (E(e itBn )) n≥1 . Since this sequence is bounded, we have proven (i).
where, by [7,Lemma 4.4], for some C(θ) > 0 depending only on θ and for Let (u n ) n≥1 and (v n ) n≥1 be two sequences of positive integers such that u n , δ −1 n /u n , v n /(δ −1 n ) and n/v n all go to infinity with n. On the one hand, where J n = min{j ≥ 1, ξ n−j+1 = 1}, (ξ j ) j≥1 being independent Bernoulli variables, ξ j having parameter θ/(θ + j − 1). On the other hand, since Ψ n (ℓ) is monotonic with respect to ℓ and tends to 1 when n/ℓ goes to infinity.

Proof of Theorem 1.1 (iii) and related statements
Since M σn has n eigenangles in each interval of length 2π, replacing X σn,δn by X ′ σn,δn changes the sum by at most quantity which, by the assumption made in (iii), tends to zero when n → ∞. Using Slutsky's lemma, we easily deduce that (i) and (ii) are preserved when we replace X σn,δn by X ′ σn,δn . In the sequel of this section, we will state alternative assumptions under which (ii) is preserved. Let us first show the following lemma: and n j=1 Ψ n (j) Proof. The equalities (6) and (7) are proven in [6,Lemma 9]. Let us now show (8). Let (u n ) n≥1 be a sequence of positive integers such that u n and n/u n both tend to infinity with n. We split the sum into two as follows n j=1 Ψ n (j) By monotonicity of Ψ n (k) with respect to k, we have Besides, which tends to 0 as n goes to infinity if we take for instance u n := ⌊max(n 1− θ 2 , n 1/2 )⌋.
Now, let us introduce the following notations: for all positive integers j and all real numbers x > 0, With this notation, all computations related to X ′ σn,δn are similar to the computations related to X σn,δn , except that Θ f and Ξ f are replaced by Θ f,ℓ and Ξ f,ℓ in the contribution of a cycle of length ℓ. We will now prove the following result: If the sequence (δ n ) n≥1 is such that where X is a Poisson point process with intensity θ x dx on (0, +∞). Proof. From Theorem 1.1, we are done if we show E(e itAn ) − E(e itCn ) −→ n→∞ 0 for all t ∈ R, where A n := n ℓ=1 a n,ℓ Ξ f 1 ℓδn and C n := n ℓ=1 a n,ℓ Ξ f,ℓ 1 ℓδn . Let t ∈ R.
Here, we use the fact that the expectation of the number of ℓ-cycles is equal to n/ℓ times the probability that 1 is in an ℓ-cycle, i.e., by the Feller coupling, E[a n,ℓ ] = n ℓ P[ξ n = ξ n−1 = · · · = ξ n+2−ℓ = 0, ξ n+1−ℓ = 1] = n ℓ We now estimate the sum over the positive indices +∞ k=⌊ℓ/2⌋+1 f k ℓδn : the sum over the negative indices behaves identically. To do this, we use the Euler-MacLaurin formula at order 2: for all positive integers p < q, and for all functions g ∈ C 2 (R), so that if g, g ′ and g ′′ are integrable at +∞, we have, letting q tend to infinity, Applying this formula to g(x) = f x ℓδn gives, with a change of variables into the integrals, where F is the antiderivative of f such that F (+∞) = 0. Then, with p = ⌊ℓ/2⌋ + 1, using Taylor-Lagrange formula at order 3 on F , at order 2 on f and at order 1 on f ′ , between 1 2δn and p ℓδn = 1 2δn Finally, using (6), (7) and (8), it follows |f ′′ (u)|du which tends to 0 as n → +∞, under the hypothesis made on f and δ n . Example 6.3.
• If f ∈ C 2 c (R) ( i.e. C 2 and compactly supported on R), then all the conditions of Theorem 6.2 are satisfied, and this for every δ n .
• If f satisfies (1), (9) and if nδ α n → 0 (it is in particular the case if δ n = n −ε for any ε ∈ 1 α , 1 ), then all the conditions of Theorem 6.2 are satisfied. Indeed, • If f ∈ S(R) ( i.e. in the Schwartz space of R), and if δ n = n −ε for any ε ∈ (0, 1), then all the conditions of Theorem 6.2 are satisfied.

Some counterexamples
• If f (x) = 1/(1+|x|), then X σn,δn (f ) is infinite, whereas in the expression of X ′ σn,δn (f ), a cycle of length ℓ gives a contribution of The sum of the lengths of the cycles larger than δ −1 n is n − o(n) with probability tending to 1 when n → ∞, and their contribution is then equivalent to 2nδ n log(δ −1 n ). The contribution of the cycles smaller than δ −1 n is dominated by δ n log(δ −1 n ) times the sum of their lengths, plus the number of these cycles. Using the Feller coupling with independent Poisson variables, one deduces that with high probability, the contribution is dominated by δ n log(δ −1 n )(δ −1 n ω(n)) + log(δ −1 n ) ≪ log(δ −1 n )ω(n) for any function ω(n) larger than 1 and going to infinity at infinity. If we take ω(n) going to infinity slower than nδ n , we deduce that X ′ σn,δn (f ) = (2 + o(1))nδ n log(δ −1 n ) with probability tending to 1 when n → ∞. This behavior at infinity does not correspond to what we get in the theorems proven earlier. If we replace X σn,δn by X ′ σn,δn , then we subtract at least n terms of the form f (x/2πδ n ) for π < x ≤ 3π, and then at least a quantity of order nδ 2 n . If nδ 2 n tends to infinity when n → ∞ (for example if δ n = n −1/3 ), then X ′ σn,δn − nδ nf (0)