Monotonous subsequences and the descent process of invariant random permutations

It is known from the work of Baik, Deift, and Johansson [1999] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutation with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting descent process of a large class of random permutations is stationary, one-dependent and determinantal.

1 Introduction and statement of results
The Tracy-Widom distribution appears in many problems of random growth, integrable probability and as the distribution of the rescaled largest eigenvalue of many models of random matrices [Corwin, 2012, Borodin andGorin, 2016]. F 2 can be expressed as the Fredholm determinant of the Airy kernel on L 2 (s, ∞), as well as in terms of the Hastings-McLeod solution of the Painlevé II equation [Tracy and Widom, 1994]. Those problems are known as a part of the Kardar-Parisi-Zhang dimension 1+1 universality class.
This work's first aim is to study the limiting behaviour of other distributions of random permutations, in particular, to prove a similar result to that of Baik, Deift and Johansson (Theorem 1). More precisely, we are interested in a class of random permutations which are stable under conjugation for which we provide a sufficient condition to obtain the Tracy-Widom fluctuations. It includes the Ewens distributions and others distributions appearing in genetics, random fragmentations and coagulation processes [Ewens, 1972, Kingman, 1975, 1978, Bertoin, 2006.
For the remainder of this article, we denote by (σ n ) n≥1 a sequence of random permutations with joint distribution P such that for all positive integer n, σ n ∈ S n . We denote by #(σ) the number of cycles of a permutation σ. For example, the identity of S n has n cycles. We prove the following.
Then for all s ∈ R, The idea of the proof we give in Subsection 3.1 is to construct a coupling between any distribution satisfying these hypotheses and the uniform distribution in order to use Theorem 1. Let us illustrate Theorem 2 with the Ewens distributions that were introduced by Ewens [1972] to describe the mutation of alleles.
Definition 3. Let θ be a non-negative real number. We say that a random permutation σ n follows the Ewens distribution with parameter θ if for all σ ∈ S n , P(σ n = σ) = θ #(σ)−1 .
Note that when θ = 1, the Ewens distribution is just the uniform distribution on S n , whereas when θ = 0 we have the uniform distribution on permutations having a unique cycle. For general θ, the Ewens distribution is clearly invariant under conjugation since it only involves the cycles' structure of θ. For our purpose, a useful property is that, if σ n follows the Ewens distribution with parameter θ > 0, then the number of cycles #(σ n ) is the sum of n independent Bernoulli random variables with parameters θ θ+i 0≤i≤n−1 . For further reading, we recommend [Aldous, 1985, McCullagh, 2011, Chafaï, Doumerc, and Malrieu, 2013. This already yields the following: Corollary 4. Let (θ n ) n≥1 be a sequence of non-negative real numbers such that: lim n→∞ θ n log(n) n 1 6 = 0. (H'2) If σ n follows the Ewens distribution with parameter θ n , then we have Tracy-Widom fluctuations (TW).
We will apply Theorem 2 for a generalized version of the Ewens distributions in Section 2. We give also other applications for random virtual permutations in Subsection 1.4.
The proof of Theorem 1 uses determinantal point processes properties obtained from the Plancherel measure which is also the law of the shape of the Robinson-Schensted correspondence of random uniform permutations. We will study in the next subsection this correspondence in the non-uniform setting and we give a more general result, see Theorem 6.

The Robinson-Schensted correspondence of random permutations
In this subsection, we study, under appropriate scalings, the limiting shape and the limiting distribution of the first components of the image of a random permutation stable under conjugation by the Robinson-Schensted correspondence.
Let n be a positive integer. A Young diagram λ = {λ i } i≥1 of size n is a partition of n i.e.
We will use the well-known application on the symmetric group S n with values in Y n known as the shape of the image of a permutation σ by the Robinson-Schensted correspondence [de B. Robinson, 1938, Schensted, 1961 or the Robinson-Schensted-Knuth correspondence [Knuth, 1970]. We denote it by We will not include here algorithmic details. For further reading, we recommend [Sagan, 2001, Chapter 3]. For our purpose, a useful property of this transform is that When σ n follows the uniform law, the distribution of λ(σ n ) on Y n is known as the Plancherel measure. In this case, after appropriate scaling, λ(σ n ) converges at the edge to the Airy ensemble. For the definition of the Airy ensemble, which is the determinantal point process associated with the Airy kernel, see for example [Tracy and Widom, 1994].
In the remainder of this paper, we denote by F 2,k (s 1 , s 2 , . . . , s k ) := P(∀i ≤ k, ξ i ≤ s i ) the cumulative distribution of the top right k particles of the Airy ensemble (ξ i ) i≥1 .
Theorem 5. [Borodin, Okounkov, and Olshanski, 2000, Theorem 5] [Johansson, 2001, Theorem 1.4] Assume that σ n follows the uniform distribution on S n . Then for all real numbers s 1 , s 2 , . . . , s k , For distributions satisfying the same assumptions as in Theorem 2, we have the same asymptotic as in the uniform setting at the edge.
Theorem 6. Assume that the sequence of random permutations (σ n ) n≥1 satisfies (H1) and (H2). Then for all positive integer k, for all real numbers s 1 , s 2 , . . . , s k , Clearly, the convergence (Ai) holds for the Ewens distributions under the hypothesis (H'2).
Using (1), Theorem 2 is a direct application of this theorem for k = 1. The proof we provide in Subsection 3.2 is a generalization of the proof of Theorem 2. We give separate proofs of Theorem 2 and Theorem 6 because the proof of Theorem 2 is simpler and do not require any knowledge of the representations of the symmetric group. Moreover, we believe that understanding the proof of Theorem 2 is helpful to understand the main idea of the proof of Theorem 6. The typical shape under the Plancherel measure was studied separately by Logan and Shepp [1977] and Vershik and Kerov [1977]. Stronger results are proved by Vershik and Kerov [1985]. In 1993, Kerov studied the limiting fluctuations but did not publish his results. See [Ivanov and Olshanski, 2002] for further details. Let L λ(σ) be the height function of λ(σ) rotated by 3π 4 and extended by the function x → |x| to obtain a function defined on R. For example, if λ(σ) = (7, 5, 2, 1, 1, 0) the associated function L λ(σ) is represented by Figure 1. For the Plancherel measure we have the following result.
Theorem 7. [Vershik and Kerov, 1985, Theorem 4] Assume that σ n follows the uniform distribution. Then for all ε > 0, where Ω(s) := 2 π (s arcsin(s) Under weaker conditions than those of Theorem 6, we show a similar result. For the remainder of this paper, we will refer to this limiting shape as the Vershik-Kerov-Logan-Shepp shape.
Theorem 8. Assume that the sequence of random permutations (σ n ) n≥1 satisfies (H1) and that for all ε > 0, Then for all ε > 0, We will prove this result in Subsection 3.2 using the same coupling as in the proof of Theorem 2.
When σ is random, D(σ) is known as the descent process.
In the non-uniform setting, the descent process is already studied for the Mallow's law with Kendall tau metric: it is also determinantal with different kernels. See [Borodin, Diaconis, and Fulman, 2010, Proposition 5.2]. Using similar techniques as in the previous subsections, we show that for a large class of random permutations, the limiting descent process is determinantal with the same kernel as the uniform setting.
Then for all finite set A ⊂ N * := {1, 2, . . . }, We will prove this result in Subsection 3.4 but before that let us illustrate it by the Ewens distributions (see Definition 3).
Corollary 11. Let (θ n ) n≥1 be a sequence of non-negative real numbers. Assume that σ n follows the Ewens distribution with parameter θ n . If lim n→∞ θ n n = 0.
Then the limiting descent process is determinantal with kernel K 0 (DPP).
Proof. Using the Chinese restaurant process interpretation of the Ewens measures, see for example [Aldous, 1985, Part II Section 11], we have P(σ n (n) = n) = θ n θ n + n − 1 .
By the stability under conjugation, We can now conclude using Theorem 10.
When θ n = 0 (the uniform measure on permutations having a unique cycle), we have a stronger result. For all positive integers n and m such that m ≥ n + 2, for all A ⊂ {1, . . . , n}, In other terms, in this case, the restriction of the descent process of σ n+2 to {1, 2, . . . , n} is determinantal with kernel K 0 . This result is a direct consequence of the main result of Elizalde [2011].

Virtual permutations
We give in this subsection another application of previous theorems. Virtual permutations are introduced by Kerov, Olshanski, and Vershik [1993] as the projective limit of S n . We are interested in this article only in random virtual permutations stable under conjugation also known as central measures as defined and totally characterized by Tsilevich [1998]. Those measures are the counterpart for random permutations of the Kingman exchangeable random partitions [Kingman, 1975[Kingman, , 1978.
We define the space of virtual permutations S ∞ as the projective limit of S n as n goes to infinity: Therefore, a random virtual permutation is a sequence (σ n ) n≥1 of random permutations such that π n (σ n+1 ) a.s = σ n . We say that it is stable under conjugation if for all positive integer n, σ n is stable under conjugation. In this case, the number of cycles can be expressed in terms of probabilities of fixed points. .

(H"2)
Then we have Tracy-Widom fluctuations (TW) and the convergence at the edge to the Airy ensemble (Ai).
Proof. By construction, for all random virtual permutation (σ n ) n≥1 and for all positive integer n, Consequently, Moreover, under the hypothesis (H"2) we have We can then conclude using Theorem 6. Similarly, using the hypothesis (H'4) we obtain: We can then conclude using Theorem 8.
According to [Tsilevich, 1998, Section 3] there exists a one-to-one correspondence between the set of probability distributions on S ∞ stable under conjugation and the set of probability distributions on Let ν be a probability measure on Σ. We denote by (σ ν n ) n≥1 a random virtual permutation stable under conjugation such that the associated distribution on Σ is ν. We will study this correspondence in three parts: Here, r j is the number of cycles of length j of σ and the sum is over all sequences of non-negative integers m = (m i ) i≥1 such that ∀j ≥ 1, |{i; m i = j}| = r j . For more details, see [Tsilevich, 1998, section 3].
Corollary 13. If x n = o(n −α ) with α > 6, then we have Tracy-Widom fluctuations (TW) and the convergence at the edge to the Airy ensemble (Ai).
We give a proof of Corollary 13 and Corollary 14 in Subsection 3.3. A trivial application of these corollaries is when x i = δ 1 (i). In this case, σ δx n follows the Ewens distribution with parameter θ = 0.
• If ν(Σ 1 ) = 1, ν is called a 1-measure. In this case, the distribution of (σ ν n ) n≥1 is a mixture of the previous distributions i.e. for all positive integer n, for all σ ∈ S n , Corollary 15. Assume that , then we have Tracy-Widom fluctuations (TW) and the convergence at the edge to the Airy ensemble (Ai).
We will prove Corollary 15 and Corollary 16 in Subsection 3.3. To explain the relation with the Ewens distributions, we need first to introduce the Poisson-Dirichlet distributions. Let θ > 0 and let . We define the random variable S := i≥1 x i . It is proved that the sum S is almost surely finite. We can find a proof for example in [Holst, 2001]. The point processx := x i S i≥1 defines a measure on Σ 1 known as the Poisson-Dirichlet distribution with parameter θ. It was introduced by Kingman [1975] and it is a useful tool to study some problems of combinatorics, analytic number theory, statistics and population genetics. See [Kingman, 1980, Donnelly and Grimmett, 1993, Arratia, Tavaré, and Barbour, 2003, Tenenbaum, 2015.
The Poisson-Dirichlet distribution with parameter θ > 0 represents also the limiting distribution of normalized cycles' lengths of the Ewens distribution with the same parameter, see [Arratia, Tavaré, and Barbour, 2003]. As a consequence, using the description of these measures in [Tsilevich, 1998, section 3], if ν follows the Poisson-Dirichlet distribution with parameter θ, σ ν n follows the Ewens measure with same parameter θ. In this case, the hypotheses of Corollaries 15 and 16 are satisfied.
• In the general case, the correspondence is given by the formula: Here, r j is the number of cycles of length j of σ and the sum is over all sequences of non-negative integers l is the number of fixed points of σ, σ j is the permutation obtained by removing j fixed points of σ and Id n is the identity of S n . For more details, we recommend [Tsilevich, 1998, section 3].
In the general case, we do not expect the Tracy-Widom fluctuations neither for ℓ nor for ℓ (see Section 2). We limit then our study to the case where there exists 0 < x 0 < 1 such that ν(Σ 1−x 0 ) = 1. Unlike all previous examples when ℓ(σ n ) and ℓ(σ n ) have the same asymptotic fluctuations, in this case, the expected length of the longest increasing subsequence is larger than (1 − x 0 )n and we will show that there exist some cases where the expected length of the longest decreasing subsequence is asymptotically proportional to √ n with Tracy-Widom fluctuations.
Corollary 17. Let 0 < x 0 < 1 and ν be a probability measure on Σ satisfying ν (Σ 1−x 0 ) = 1. Letν be the 1-measure such that dν(x) = dν x 1−x 0 . If there exists a positive integer k such that for all real numbers s 1 , s 2 , . . . , s k , then for all real numbers s 1 , s 2 , . . . , s k , In particular, for all real s, This corollary is a direct application of Proposition 21. Here is some examples of measures ν that meet the assumptions of the previous corollary: In fact: satisfies hypotheses of Corollary 13.
For the descent process, we have the following result: The proof of this result we suggest in Subsection 3.4 consists in studying in a first step the case where the corresponding measure ν is concentrated on Σ 1 . We prove that the limiting point process is determinantal with kernel (i, j) → k 0 (j − i). In a second step, we prove that the kernel depends only on i≥1 x i .
Theorem 18 implies that for a general random virtual permutation stable under conjugation, we have the following result.
Corollary 19. For any probability measure ν on Σ, For the total number of descents we have Proposition 20. For any probability measure ν on Σ, We will prove Corollary 19 and Proposition 20 in Subsection 3.4.

Further discussion
In previous subsections, except for Corollary 4, the applications are for virtual permutations, but with the same logic, we can prove a similar result as Corollary 17 for some permutations non compatible with projections.
Proposition 21. Let (P n ) n≥1 be a sequence of probability measures stable under conjugation. Assume that there exists a positive integer k such that for all real numbers s 1 , s 2 , . . . , s k , Let 0 ≤ x 0 < 1 and (σ n ) n≥1 be a sequence of random permutations such that for all positive integer n, for all σ ∈ S n , where l is the number of fixed points of σ and σ j is the permutation obtained by removing j fixed points of σ.
Then for all real numbers s 1 , s 2 , . . . , s k , We prove this result in Subsection 3.3. An interpretation of the random permutation defined by equation (9) is the following. Let n be a positive integer. We construct a subset A of {1, 2, . . . , n} as follows: for every 1 ≤ i ≤ n, with probability x 0 , i ∈ A independently from other points. The points of A are then fixed points of σ n . After that, we permute the elements of {1, 2, . . . , n} \ A according to the probability distribution P n−|A| . In particular, A is a subset of all fixed points of σ n .
As a consequence, recalling (5), if there exists 0 < x 0 < 1 such that ν (Σ 1−x 0 ) = 1, then the number of fixed points of σ ν n is larger than a binomial random variable with parameters x 0 and n. Consequently, In this case, we conjecture that the fluctuations are Gaussian.
A possible generalization of the Ewens distributions is the following.
Definition 23. Letθ = (θ i ) i≥1 be a sequence of positive real numbers, we say that σ n follows the generalized Ewens distribution on S n with parameterθ if for all σ ∈ S n , Here, r i (σ) is the number of cycles of σ of length i.
This generalization was studied in some cases in details by Ercolani and Ueltschi [2014]. In the general case, it is not obvious to have a good control on the number of cycles. Nevertheless, by using some results of Ercolani and Ueltschi, we can conclude in some cases.
Corollary 24. Let (σ n ) n≥1 be a sequence of random permutations such that for all positive integer n, σ n follows the generalized Ewens distribution with parameterθ = (θ i ) i≥1 . Assume thatθ satisfies one of the following hypotheses: Then we have Tracy-Widom fluctuations (TW) and the convergence at the edge to the Airy ensemble (Ai).
For the descent process, we have the convergence for a larger class of parameters.
Corollary 25. Let (σ n ) n≥1 be a sequence of random permutations such that for all positive integer n, σ n follows the generalized Ewens distribution with parameterθ = (θ i ) i≥1 . Assume thatθ meets one of the hypotheses of the previous corollary or lim i→∞θ i i γ = 1, where γ ≥ 0. We have then the convergence of D(σ n ) to the determinantal point process with kernel K 0 (DPP).
Corollaries 24 and 25 are a direct application from the computations of Ercolani and Ueltschi. In particular, we use the following results: Lemma 26. Letθ = {θ i } i≥1 and {σ n } n≥1 be a sequence of random permutations following the generalized Ewens distribution with parameterθ.
• Ifθ i → θ, then 1 θ log(n) E(#(σ n )) → 1 [Ercolani and Ueltschi, 2014, Theorem 6.1]. Using this lemma, it is obvious that (H2) is satisfied under the assumptions of Corollary 24. Moreover, (H4) can be replaced by This result is a consequence of the stability under conjugation. Indeed, Using this observation, it is obvious that (H4) is satisfied under assumptions of Corollary 25.
Pitman [1992] introduced a two-parameters generalization of the Ewens distribution. Using the same notations as in [Pitman, 1992], we can apply Theorems 6 for α < 1 6 and Theorem 8 for α < 1.
The bound n 1 6 of Theorem 2 may not be optimal. The best counterexample we found is when the number of cycles is of order √ n for the general case and of order n for virtual random permutations. Nevertheless, using the same lines of proof, we can obtain the convergence of ℓ(σn) √ n with optimal hypotheses. Proposition 27. Assume that the sequence of random permutations (σ n ) n≥1 satisfies (H1) and the number of cycles is such that: For all ε > 0, In this case, the bound √ n in the second condition is optimal.
Knowing that all elements of S n with a unique cycle belong to the same class of conjugation, they are equally distributed and Lemme 29 follows from (11).
The previous Lemma is equivalent to say that T n−1 (σ n ) follows the Ewens distribution on S n with parameter θ = 0.
Using Lemma 29, T n−1 (σ n ) does not depend on the law of σ n . Therefore, it is enough to prove Theorem 2 for one particular case. In fact, the convergence (TW) has been obtained for the uniform setting, see Theorem 1. By choosing (σ n ) n≥1 a sequence of random permutations following the uniform distribution, we have then (TW) for the Ewens distribution with parameter θ = 0. For the general case, if the sequence (σ n ) n≥1 satisfies (H1) and (H2), we can conclude using Lemma 29 and (12).
The same argument can be applied for the length of longest decreasing subsequence.

Proof of results related to the Robinson-Schensted transform of random permutations
To prove Theorems 6 and 8 we need to recall a well-known property of the Robinson-Schensted correspondence. Let σ ∈ S n . We denote

We have then
Lemma 30. [Greene, 1974] For any permutation σ ∈ S n , In particular, max This result is proved first by Greene [1974] (see also [Sagan, 2001, Theorem 3.7.3]). It will be the keystone to prove Theorem 6 and Theorem 8 as it implies the following lemma which is the counterpart of Lemma 28.
Lemma 31. For any permutation σ and transposition τ , Moreover, Proof. Let σ be a permutation and τ = (l, m) be a transposition. We have then for all integer i, and similarly Consequently, by Lemma 30, Using the same argument with σ • τ instead of σ, (13) follows. Moreover, since the triangle inequality yields (14).
Proof of Theorem 6. Similarly to the proof of Theorem 2, we will use the same Markov operator T to compare our random permutation with the uniform distribution. Using Lemma 31 and the equality (10) we obtain Consequently, under (H2), ∀ε > 0, The remaining of the proof is similar to the proof of Theorem 2.
We will now prove Theorem 8.
Lemma 32. Let α, β ∈ N and A the point such that We have also the following result.
Proof. Let M such that − − → OM = s 1 u + s 2 v. By construction, if M ∈ C λ then s 1 , s 2 ≥ 0 and either s 1 ∈ N or s 2 ∈ N. If we apply this observation to M defined by we obtain (18).
To prove Theorem 8, our main lemma is the following.
Proof. Note that for any i ∈ Z, s → L λ (s) and s → L µ (s) are affine functions on √ 2 2 i, √ 2 2 (i + 1) and thus (19) is equivalent to Let i ∈ Z. It follows from Lemma 33 that there exists k i ∈ Z such that, To simplify notations, we denote Let A and B be the points such that Clearly A ∈ C λ and B ∈ C µ . By Lemma 33, i+j 2 , j−i 2 ∈ N. We can then apply Lemma 32. In the case where k i > 0, we have Using the fact that (λ l ) l≥1 and of (µ l ) l≥1 are decreasing, we have, Similarly, in the case where k i < 0,

Proofs of the applications to virtual permutations
We will prove in this subsection Corollaries 13, 14, 15 and 16 and Proposition 21. We will not give details of the proof of Corollary 17 because it is a direct application of Proposition 21.
We can have a combinatorial interpretation of (5). Let x = (x i ) i≥1 ∈ Σ. At the beginning, we have an infinite number of circles {C n } n∈Z . At each step n ≥ 1 we choose an integer pos n with probability distribution j≥1 x j δ j + (1 − i≥1 x i )δ 0 independently from the past. We insert then the number n uniformly on the circle C posn if pos n > 0 and on the circle C −n if pos n = 0 . At each step, one reads the elements on each non-empty circle counterclockwise to get a cycle. For example, if pos 1 = 4, pos 2 = 1, pos 3 = 4, pos 4 = 0 and pos 5 = 0, we obtain the permutation (1, 3)(2)(4)(5). With this description, we have Proof of Corollary 13 and Corollary 14. In both corollaries, since i≥1 x i = 1, we have If α > 6, there exists a real number β such that 5 6(α−1) < β < 1 6 . Moreover there exists n 0 such that ∀n > n 0 , x n < n −α . For any n > (n 0 ) 1 β and under hypothesis of Corollary 13, we have Then Corollary 13 follows from Theorem 6. If α > 1 and under hypothesis of Corollary 14, there exists n 0 such that ∀n > n 0 , x n < n −α and let n > (n 0 ) Then Corollary 14 follows from Theorem 8.
Proof of Corollary 15 and Corollary 16.
Then, we obtain Corollary 15 and Corollary 16 thanks to Theorem 6 and Theorem 8.
Proof of Proposition 21. An interpretation of the random permutation defined by equation (9) is the following. Let n be a positive integer. We construct a subset A n of {1, 2, . . . , n} as follows: for every 1 ≤ i ≤ n, with probability x 0 , i ∈ A n independently from other points. The points of A n are then fixed points of σ n . After that, we permute the elements of {1, 2, . . . , n} \ A n according to the probability distribution P n−|An| .
The main idea is that a decreasing subsequence cannot have more than one element belonging to A n . Moreover, a decreasing subsequence of the restriction of σ n on {1, 2, . . . , n} \ A n is a decreasing subsequence of σ n . In other words, for all real number s, for all 1 ≤ j ≤ n, More generally, using Lemma 30, we have for all real numbers s 1 , . . . , s k , Consequently, In the sequel of the proof, let s 1 , . . . , s k be k real numbers and ε > 0. As |A n | is a random binomial variable with parameters n and x 0 , and using the central limit theorem, there exist n 0 , α > 0 such that, n 0 > α 2 (1−x 0 ) 2 and ∀n > n 0 , We denote by p n j := P(|A n | = n − j),x 0 := 1 − x 0 ,k = 2k − 1. As and Here, ⌊x⌋ and ⌈x⌉ are respectively the floor and the ceiling functions.

Proof of results for the descent process
In this subsection, we prove the convergence of the descent process for some random permutations stable under conjugation (Theorem 10). We prove also results of convergence for virtual permutations (Theorem 18, Corollary 19 and Proposition 20).
Let A be a finite subset of N * and m := max(A) and let A ′ = {1, 2, . . . , m + 1}. The idea of the proof of Theorem 10 is to study the descent process under the condition {σ n (A ′ ) ∩ A ′ = ∅} and to show that it does not depend on the law of σ n .
Proof of Theorem 10. Under (H4), Similarly, ifσ n follows the uniform distribution on S n , we have P(σ n ∈ E n ) → 1. Therefore, since the law of σ n is invariant under conjugation (H1) we can use Lemma 35 for n large enough to get Thus, lim n→∞ (P(A ⊂ D(σ n )) − P(A ⊂ D(σ n ))) = 0.
Sinceσ n satisfies (DPP) by Theorem 9, this concludes the proof.
Before proving Theorem 18, we need to recall that a point process X on a discrete space X is fully characterised by its correlation function (we denote it by ρ). Given A a finite subset of X, It is called determinantal with kernel K if for all A finite subset of X, A point process defined on N * is 1-dependent if for all A and B finite subsets of N * such that the distance between A and B is larger than 1, ρ(A ∩ B) = ρ(A)ρ(B). It is called stationary on N * if for all positive integer k, for all finite subset A ⊂ N * , ρ(A) = ρ(A + k).
To prove Theorem 18, we will use the following result.
Proof of Theorem 18. If x 0 = 1, the theorem is obvious since D(σ ν n ) = δ ∅ . Next we split the proof into two steps depending on whether x 0 = 0 or not.
Using the same combinatorial interpretation as in the beginning of Subsection 3.3, we have for any x ∈ Σ 1 , As for all i ≤ n 0 , x i (1 − x i ) n−1 converges to 0 when n goes to infinity, there exists n 1 such that for n > n 1 n 0 i=1 x i (1 − x i ) n−1 < ε 2 and therefore P(σ δx n (1) = 1) → 0.
Step 2: we now assume that 0 < x 0 < 1 and ν(Σ 1−x 0 ) = 1. We have which prevents the use of Theorem 10. The strategy is instead to use Theorem 36, namely to prove that the limiting process is stationary, 1-dependent and its correlation function is such that ∀k ≥ 1, To do so we need to prove this result in the particular case dν 1 (x) := dP D(1)( x 1−x 0 ) since for any finite subset B, lim n→∞ (P(B ⊂ D (σ ν n )) − P(B ⊂ D(σ ν 1 n ))) = 0.
Indeed, let B be a finite subset of N * and B ′ := B ∪(B +1). We use the same interpretation of the random virtual permutations in this case as in the proof of Proposition 21. We choose a random subset A n of {1, 2, . . . , n} of fixed points where each point belongs to A n with probability x 0 independently from the others. After that, we permute the elements according to P n−|An| , where (P n ) n≥1 is the probability distribution on S ∞ associated tô ν where dν(x) = dν( x 1−x 0 ). Let C n := A n ∩ B ′ and E n := {σ ∈ S n , ∀i ∈ B ′ \ C n , σ(i) > max(B ′ )}.

We have
With similar arguments as in the proof of Lemma 35, it is not difficult to show that the quantity P(B ⊂ D (σ ν n ) |E n , C n = X) is defined for n > |B ′ | + max(B ′ ) and does not depend on ν. Moreover, Moreover, using the notation p k := P(σν k (1) = 1) and observing that p k → ∞ as k → ∞ thanks to Step 1, we have This yields lim n→∞ P(σ ν n ∈ E n ) = 1 and therefore the claim (30) is proven.
The finite subset B can be decomposed as B = l i=1 B i where each B i consists in consecutive elements of N * and the distance between B i and B j is larger than one if i = j. Note that every finite subset has a such decomposition. Let B ′ i := B i ∪ (B i + 1). We have B ′ := B ∪ (B + 1) = l i=1 B ′ i and if i = j, then B ′ i ∩ B ′ j = ∅. From now we assume that n > |B ′ | + max(B ′ ). We have P(B ⊂ D(σ ν 1 n )|E n ) = X⊂B ′ P(B ⊂ D(σ ν 1 n )|C n = X, E n )P(C n = X).
Corollary 19 is at the same time a generalization and a direct application of Theorem 18.
Proof of Corollary 19. We denote by f (n, x, σ) := P σ δx n = σ (see (5)), by ρ(n, x, .) the correlation function of the descent process of σ δx n and by ρ lim (x 0 , .) the correlation function of the determinantal process with kernel K x 0 (i, j) := k x 0 (j − i). Let A be a finite subset of N * . We have Using the convergence of ρ(n, x, A) to ρ lim (1 − i≥1 x i , A) and the dominated convergence theorem, we obtain: Using this corollary, we can now proof Proposition 20.
Lemma 37. For any random permutation σ n stable under conjugation, P(i ∈ D(σ n )) does not depend on i.
Proof of Proposition 20. Let ν be a probability measure on Σ. By Lemma 37 and by using (7) and (8)