Salma Ben Dhaou
email: salma.bendhaou@hotmail.fr

Mouloud Kharoune
email: mouloud.kharoune@univ-rennes1.fr

Boutheina Arnaud Martin
email: arnaud.martin@univ-rennes1.fr

Ben Yaghlane
email: boutheina.yaghlane@ihec.rnu.tn

Boutheina Ben

Salma Ben

Dhaou Larodec

Belief Approach for Social Networks

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Social networks appeared long before the birth of Internet. A social network can be defined as a group of persons or organizations connected between them by relations and social exchanges which they maintain. However, with the evolution of connection rates and collaborative technologies which are continuously changing, Internet provides access to new networks that are wider, and more playful social but also less easily recognizable.

Furthermore, an important volume of incomplete and imperfect information are spreading on the network. Therefore, the management of the uncertainty is fundamental in several domains, especially in social networks. In fact, belief functions theory allows, not only the representation of the partial knowledge, but also the fusion of information. In the case of social networks, this theory allows to attribute mass functions to the nodes which represent, for example, persons, associations, companies and places as well as links that can be friendly, family and professional and on messages that can be of type for example: personal commercial, personal not commercial, impersonal commercial and impersonal not commercial. Therefore, we will have a global view on exchanges made on the network and this will lead us to make a better decision.

In addition, by using uncertainty, we can better monitor the behaviour of the social network. Thus, extending the work on the real plane, we can predict such a terrorist act or assess the quality of a product or follow a buzz. . . In this context, previous works have focused on models and methods devoted to the analysis of social network data [START_REF] Wasserman | Social Network Analysis, Methods and Application. The press syndicate of the university of cambridge[END_REF] [2] while others have interested in information fusion in order to have a global information about the network [START_REF] El Zoghby | Fusion distribuée évidentielle pour la détection d'attaque sybil dans un réseau de véhicules (1)[END_REF].

The aim of this paper is to propose a new model, a belief social network which is a network supplied by the masses. In fact, we attribute a mass function to the nodes, edges and messages.

This paper is structured as follows. In section 2, we briefly recall some concepts related to the theory of belief functions. We propose in section 3 our model: the belief social network. In section 4, we present the fusion of the masses on belief social network. Finally, section 5 is devoted to illustrate the belief social network and section 6 concludes the paper.

Basic concepts of belief functions

In this section, we will remind the basic concepts of the theory of belief functions used to instrument our model, the belief social network. Let Ω be a finite and exhaustive set whose elements are mutually exclusive, Ω is called a frame of discernment. A mass function is a mapping m : 2 Ω → [0, 1] such that ∑ X∈2 Ω m(X) = 1 and m( / 0) = 0. The mass m(X) expresses the amount of belief that is allocated to the subset X. In order to deal with the case of the open world where decisions are not exhaustive, Smets [START_REF] Smets | Belief Functions: the Disjunctive Rule of Combination and the Generalized Bayesian Theorem[END_REF] proposed the conjunctive combination rule. This rule assumes that all sources are reliable and consistent. Considering two mass functions m 1 and m 2 for all A ∈ 2 Ω , this rule is defined by:

m ∩ (A) = ∑ B∩C=A m 1 (B) * m 2 (C) (1)
We will also consider the normalized conjunctive rule, the Dempster rule, given for two mass functions m 1 and m 2 for all x ∈ 2 Ω by:

m ⊕ (A) = m ∩ (A) 1 -m ∩ ( / 0) (2)
The coarsening corresponds to a grouping together the events of a frame of discernment Θ to another frame compatible but which is more larger Ω [START_REF] Smets | Belief Functions: the Disjunctive Rule of Combination and the Generalized Bayesian Theorem[END_REF][START_REF] Smets | Imperfect information: Imprecision -Uncertainty[END_REF]. Let Ω and Θ be two finite sets. The refinement allows the obtaining of one frame of discernment Ω from the set Θ by splitting some or all of its events [START_REF] Shafer | A mathematical theory of evidence[END_REF].

In order to make a decision, we try to select the most likely hypothesis which may be difficult to realize directly with the basics of the theory of belief functions where mass functions are given not only to singletons but also to subsets of hypothesis. Some solutions exist to ensure the decision making within the theory of belief functions. The best known is the pignistic probability proposed by the Transferable Belief Model (TBM). Other criteria exists like the maximum of credibility and the maximum of plausibility [START_REF] Janez | Fusion de sources d'information définies sur des référentiels non exhaustifs différents[END_REF].

The TBM is based on two level mental models: The "credal level" where beliefs are entertained and represented by belief function and the "pignistic level" where beliefs are used to make decision and represented by probability functions called the pignistic probabilities. When a decision must be made, beliefs held at the credal level induce a probability measure at the pignistic measure denoted BetP [START_REF] Smets | Constructing the pignistic probability function in a context of uncertainty[END_REF]. The link between these two functions is achieved by:

BetP(A) = ∑ B⊆Θ |A ∩ B| |B| m(B) 1 -m( / 0) , ∀A ⊆ Θ (3)
To focus on the type of relationship between two different frames of discernment, we may use the multi-valued mapping introduced by Hyun Lee [START_REF] Lee | Context reasoning under uncertainty based on evidential fusion networks in home based care[END_REF]:

m Γ (B j ) = ∑ Γ (e i )=B j m(e i ) (4)
with e i ∈ Ω and B j ⊆ Θ . Therefore the function Γ is defined as follow Γ : Ω → 2 Θ . The vacuous extension, being a particular case of multi-valued mapping has the objective to transfer the basic belief assignment of a frame of discernment Ω towards the Cartesian product of frames of discernment Ω × Θ . The operation of vacuous extension, noted ↑, is defined by:

m Ω ↑Ω ×Θ (B) = m Ω (A) if B = A ×Θ 0 otherwise (5)
The marginalization allows, from a basic belief assignment defined on a space produced to find the basic belief assignment on one of the frames of discernment of the produced space. This operation, noted ↓ is defined by:

m Ω ×Θ ↓Ω (A) = ∑ B⊆Ω ×Θ m Ω ×Θ (B) ∀A ⊆ Ω ( 6
)
where A is the result of the projection of B on Ω .

Belief Social Network

Several works have focused on the representation of networks with graphs. A classical graph is represented by G = {V ; E} with: V a set of type's nodes and E a set of type's edges. This representation does not take into account the uncertainty of the nodes and edges.

In fact, graphical models combine the graph theory with any theory dealing with uncertainty like probability [START_REF] Parchas | The pursuit of a good possibe world: Extracting representative instances of uncertain graphs[END_REF], [START_REF] Khan | Fast reliability search in uncertain graphs[END_REF] or possibility or theory of belief functions to provide a general framework for an intuitive and a clear graphical representation of real-world problems [START_REF] Laamari | Dynamic directed evidential networks with conditional belief functions:application to system reliability[END_REF]. The propagation of messages in networks has been modelled using the theory of belief functions combined with other theories such as hidden Markov chains [START_REF] Ramasso | Contribution of belief functions to hidden markov models with an application to fault diagnosis[END_REF].

In this context, we introduce our model: the belief social network which has the role of representing a social network using the theory of belief functions. Indeed, we will associate to each node, link and message an a priori mass and observe the interaction in the network to determine the mass of the message obtained in a welldefined node. To do this, we consider an evidential graph G = {V b ; E b } with: V b a set of nodes and E b a set of edges. We attribute to every node i of V b a mass m Ω N i defined on the frame of discernment Ω N of the nodes. Moreover, we attribute also to every edge (i, j) of E b a mass m Ω L i j defined on the frame of discernment Ω L of the edges. Therefore, we have:

V b = {V i , m Ω N i } (7)
and

E b = {(V b i ,V b j ), m Ω L i j } (8)
This evidential graph structure is given by Fig 1 . In social network, we can have for example the frame of the nodes given by the classes Person, Company, Association and Place. The frame of discernment of the edges can be Friendly, Professional or Family. Moreover we note: Ω N = {ω n 1 , . . . , ω n N } and Ω L = {ω l 1 , . . . , ω n L }.

In social network, many messages can transit in the network. They can be categorized as commercial, personal, and so on.The class of the message is also full of uncertainty. Therefore to each message, we add a mass function in the considered frame of discernment

Ω Mess = {ω M 1 , . . . , ω M k }.

Fusion of masses on belief social network

In social network, we can receive the same information from different users. But, can we have the confidence to this information? Moreover, the information can be contradictory. We propose here to take into account the structure of belief social network presented in the previous section to analyse the messages received by one node. In order to integrate the belief on the nodes and on the edges, we first make a vacuous extension on Ω N × Ω L for each mass for the nodes of V b and on each mass for the edge of E b . Therefore, we obtain on each node V b i a mass: m Ω N ×Ω L i and on each edge

E i j = (V b i ,V b j ) between the nodes V b i and V b j a mass: m Ω N ×Ω L i j
. If we consider a coming message from the node V b i to the node V b j through the edge E i j , the belief of the network m Ω N ×Ω L R is given by the mass function on the node V i and the edge E i j :

m

Ω N ×Ω L R = m Ω N ↑Ω N ×Ω L V i ⊕ m Ω L ↑Ω N ×Ω L E i j (9)
Here, the index R denotes the resulted belief network from the nodes and the link between them regardless of the message.

We use the multi-valued operation to combine mass functions on different frames of discernment. In fact, a multi-valued mapping Γ describes a mapping function:

Γ : Ω N × Ω L → Ω Mess ( 10
)
We can calculate these equations by using the formula (4):

Γ : m Ω Mess Γ (B j ) = ∑ Γ (e i )=B j m Ω N ×Ω L (e i ) (11)
with e i ∈ Ω N × Ω L and B j ⊆ Ω Mess . From the function Γ , we can combine the mass given by the network m Ω Mess Γ and the mass of the message to obtain the mass of the message considering the network:

m Ω Mess R = m Ω Mess ∩ m Ω Mess Γ (12)
Now, if we consider n messages coming from n different nodes V b i 1 , . . . ,V b i n to the same node V b j . We can merge the obtained results from the equation ( 12) for the n nodes. The goal is to obtain a more precise information on an event describe by the n messages. We then take into account the local network m Ω Mess R i of the node V b j . A local network is defined as a branch of the global network composed of many nodes linked to a same node, the connecting links and the received messages. For example, we can have two nodes which send two messages simultaneously to a third node. We obtain the mass of the global network m Ω Mess G R :

m Ω Mess G R = m Ω Mess R 1 ∩ m Ω Mess R 2 ∩ . . . ∩ m Ω Mess R n (13)
Then, we will be able to take a decision on the nature of the resulting message with the pignistic probability using equation (3).

Illustrations

In this section, we will present various experiments conducted to validate our model. We consider three frames of discernment of the nodes, the links and the messages: Ω N = {Person,Company, Association, Place}, Ω L = {Friendly, Family, Pro f .}, (Pro f . for professional), Ω Mess = {PC, PNC, IC, INC}, with PC for Personal Commercial, PNC for Personal Not Commercial, IC for Impersonal Commercial and INC for Impersonal Not Commercial. We used the passage function Γ given in Table 1 which allows.

Γ Person Association Company Place Friendly PNC PNC ∪ INC PC ∪ IC INC ∪ IC Family PNC ∪ INC PNC ∪ INC PC ∪ IC INC ∪ IC Professional PNC ∪ IC IC IC IC
Table 1: Definition of the function Γ given the correspondences between Ω N × Ω L and Ω Mess .

For the purposes of our model, we will evaluate three cases. For the first one, we consider a mass function associated to: Following our proposed procedure, first, we calculate the vacuous extension of m Ω N and m Ω L on Ω N × Ω L and we combine both mass functions. We obtain:

m Ω N ×Ω L R ({Person, Friendly}) = 0.5625 m Ω N ×Ω L R ({Person, Friendly}, {Person, Family}, {Person, Pro f .}) = 0.1875 m Ω N ×Ω L R ({Person, Friendly}, {Association, Friendly}, {Company, Friendly}, {Place, Friendly}) = 0.1875 m Ω N ×Ω L R (Ω N × Ω L ) = 0.0625 (14)
Then, we use the Γ function to calculate the passage from Ω N × Ω L to Ω Mess . We obtain:

m Ω Mess Γ (PNC) = 0.5625 m Ω Mess Γ (Ω Mess ) = 0.4375 (15)
Then, we make the conjunctive combination of m Ω Mess Γ and m Ω Mess :

m Ω Mess R (PNC) = 0.8250 m Ω Mess R (Ω Mess ) = 0.1750 (16)
Finally, to make a decision, we calculate the pignistic probability:

BetP(PC) = 0.0438 BetP(IC) = 0.0438 BetP(PNC) = 0.8687 BetP(INC) = 0.0438 (17)
If we consider the results, we note that the pignistic probability on Personal Not Commercial is 0.8687. This pignistic probability was equal to 0.7 before considered the network. Hence, we show that considering the network we can reinforce our belief for a given message.

In the second case, we consider the same network, with the same masses m Ω N and m Ω L , but we consider a mass function associated to a message with: given by equation ( 14) as before and also using the Γ function the same mass given by the equation (15).

m Ω Mess
However the result of the conjunctive combination m Ω Mess Γ and m Ω Mess is now:

m Ω Mess R ( / 0) = 0.3375 m Ω Mess R (PC) = 0.2625 m Ω Mess R (PNC) = 0.2250 m Ω Mess R (Ω Mess ) = 0.1750 (18)
In this case there is a conflict between the information of the network and the message, therefore a mass come out the empty set. The pignistic probability gives:

BetP(PC) = 0.4623 BetP(IC) = 0.0660 BetP(PNC) = 0.4057 BetP(INC) = 0.0660 (19)
We note that in the first example, the highest pignistic probability is associated with the Personal Not Commercial message that had the larger mass function at the beginning. While in the second example, we find ourselves faced with almost equal probability of Personal Not Commercial and Personal Commercial types where the need for a second decision on the type of message received. Now we consider the fusion of the two examples cited above that come on the same node. We obtain the results given in Table 2. We note that by combining the two examples, we get the message Personal Not Commercial that has the highest pignistic probability. Working on real data, we can assign the mass functions to the nodes, edges and messages by evaluating certain parameters, for example, the type of contacts that are related to the profile in question as well as the type of publications produced (case of facebook).

Conclusion

In this work we presented in the first section a general introduction in which we reviewed the notion of social networks and the interest of the proposed method to respond to the expectations for reasoning under uncertainty. In the second section, we briefly introduced the basic concepts used in the theory of belief functions. Then we focused on the introduction of our model and the different notation used. Indeed, we treated step by step development of the construction of the graph. Finally, we detailed the process of merging the information flowing through the network. We also showed how the process is carried out of the fusion and explained how we can make a decision on the nature of the messages received by using the pignistic probability. In fact, in many cases, we can take a new decision on the nature of the message received by a well-defined node. This idea was explained in the second example in the illustration part. In future work, we aim to represent the update of the elements composing the network as well as to scale.

Fig. 1 :

 1 Fig. 1: Evidential graph structure for social network.

• 1 (

 1 a node with: m Ω N (Person) = 0.75 and m Ω N (Ω N ) = 0.25 • a link with: m Ω L (Friendly) = 0.75 and m Ω L (Ω L ) = 0.25 • a message with: m Ω Mess PNC) = 0.6 and m Ω Mess 1 (Ω Mess ) = 0.4

2 (

 2 PC) = 0.6 and m Ω Mess 2 (Ω Mess ) = 0.4 In this case the mass is on the Personal Commercial instead of Personal Non Commercial. As the network is the same we obtain the same mass m Ω N ×Ω L R

Table 2 :

 2 Fusion of the two examples: the mass function and the pignistic probability

	Focal Mass	Message BetP
	/ 0 0.5541	PC	0.1202
	PNC 0.3694	IC	0.0172
	PC 0.0459	PNC	0.8455
	Ω Mess 0.0306	INC	0.0172


Julien Reygner
email: julien.reygner@polytechnique.org

Chaoticity of the stationary distribution of rank-based interacting diffusions

Keywords: Mathematics Subject Classification. 60H10, 60F05 Rank-based interacting diffusions, nonlinear diffusion process, stationary distribution, chaoticity, Wasserstein distance

Chaoticity of the stationary distribution of rank-based interacting diffusions

For all n ≥ 1, consider the system of rank-based interacting diffusions, or particles, [START_REF] Banner | Atlas models of equity markets[END_REF] ∀i ∈ {1, . . . , n}, By the Girsanov theorem, the stochastic differential equation (1) possesses a unique weak solution, and actually a unique strong solution, see [START_REF] Veretennikov | Strong solutions and explicit formulas for solutions of stochastic integral equations[END_REF] for instance. When the number n of particles grows to infinity, propagation of chaos results toward the unique weak solution to the nonlinear (in McKean's sense) stochastic differential equation ( 2) dX(t) = b(F t (X(t)))dt + σdW (t),

dX n i (t) = b n   n j=1 1 {X n j (t)≤X n i (t)}   dt + σdW i (t),
F t (x) = P(X(t) ≤ x),
where obtained in [START_REF] Jourdain | Diffusion processes associated with nonlinear evolution equations for signed measures[END_REF][START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF]; we also refer to [START_REF] Jourdain | Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation[END_REF][START_REF] Dembo | Large deviations for diffusions interacting through their ranks[END_REF] for nonconstant diffusion coefficients. These propagation of chaos results are not uniform in time, and therefore do not provide any indication on the link between the long time behaviour of the particle system, which was studied in [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF][START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF], and the long time behaviour of the nonlinear diffusion process, which was described in [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF][START_REF] Jourdain | Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation[END_REF]. The purpose of this note is to clarify this link by showing that the stationary distribution of a suitably modified version of the particle system is chaotic with respect to the stationary distribution of the nonlinear process. 1.2. Projected particle system and nonlinear diffusion process. As was remarked in [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF], the solution to (1) cannot converge to an equilibrium, since its projection along the direction (1, . . . , 1) is a drifted Brownian motion. One can however address the long time behaviour of the projection onto the hyperplane which is orthogonal to the singular direction (1, . . . , 1). The resulting process is called the projected particle system, it is the M n -valued diffusion process solving

dZ n i (t) =   b n   n j=1 1 {Z n j (t)≤Z n i (t)}   - b  dt + σ n -1 n dW i (t) - σ n j =i dW j (t),
where b := 1 n n k=1 b n (k) = B [START_REF] Banner | Atlas models of equity markets[END_REF]. Propagation of chaos for the projected particle system toward the nonlinear diffusion process [START_REF] Chatterjee | A phase transition behavior for Brownian motions interacting through their ranks[END_REF] was established in [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF]. The long time behaviour of the projected particle system was addressed by [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF][START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF], under the following natural assumption:

(E) b = B(1) = 0, and the function b is decreasing on [0, 1]. The following proposition is due to Pal and Pitman [10, Theorem 8]; see also Jourdain and Malrieu [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF]Theorem 2.12]. We use the notation z (1) ≤ • • • ≤ z (n) to refer to the order statistics of a vector (z 1 , . . . , z n ) ∈ R n . Proposition 1. Under Assumption (E), for all n ≥ 1,

Z n := z∈Mn exp 2 σ 2 n k=1 b n (k)z (k) dz < +∞,
and the probability distribution with density

p n ∞ (z) := 1 Z n exp 2 σ 2 n k=1 b n (k)z (k)
with respect to the surface measure dz on M n is the unique stationary distribution of the process (Z n 1 (t), . . . , Z n n (t)) t≥0 . Let us remark that the density p n ∞ (z) only depends on the order statistics of z, and therefore is invariant under the permutations of the coordinates of z. As a consequence, the probability distribution P n ∞ := p n ∞ (z)dz is a symmetric probability distribution on R n , which gives full measure to M n .

On the other hand, the stationary distributions of the nonlinear diffusion process were described in [START_REF] Jourdain | Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation[END_REF]. This description relies on the function Φ introduced in Lemma 2 below.

Lemma 2. Under Assumption (E),

(i) b(0) > 0 > b(1) and, for all u ∈ (0, 1), B(u) > 0, (ii) the function Φ : (0, 1) → R defined by

(3) ∀u ∈ (0, 1), Φ(u) := u v=0 vσ 2 2B(v) dv - 1 v=u (1 -v)σ 2 2B(v) dv
is C 2 and increasing on (0, 1), and satisfies

(4) Φ(u) ∼ σ 2 2b(0) log(u) when u ↓ 0, Φ(u) ∼ σ 2 2b(1) log(1 -u) when u ↑ 1.
Besides, it is integrable on [0, 1] and such that (5)

1 u=0 Φ(u)du = 0.
Proof. Under Assumption (E), the chain of inequalities b(0

) > 0 > b(1) is straightforward. Besides, for all u ∈ (0, 1), 1 u u v=0 b(v)dv > b(u) > 1 1 -u
therefore the integrals in the right-hand side of (3) are finite, and the function Φ is C 2 and increasing on (0, 1), and satisfies [START_REF] Fernholz | Stochastic portfolio theory[END_REF]. The integrability of Φ on [0, 1] follows from (4) and the continuity of Φ on (0, 1), and by the Fubini-Tonelli theorem,

1 u=0 u v=0 vσ 2 2B(v) dvdu = 1 v=0 v(1 -v)σ 2 2B(v) dv = 1 u=0 1 v=u (1 -v)σ 2 2B(v) dvdu,
whence [START_REF] Fernholz | Stochastic portfolio theory: A survey[END_REF].

Note that the inverse function Φ -1 of the function Φ defined in Lemma 2 is the cumulative distribution function F ∞ of a probability distribution P ∞ on R, which is such that

x∈R |x|P ∞ (dx) = 1 u=0 |Φ(u)|du < +∞, x∈R xP ∞ (dx) = 1 u=0 Φ(u)du = 0.
Besides, since Φ is C 2 and Φ ′ (u) > 0 for all u ∈ (0, 1), we deduce that P ∞ possesses a density p ∞ with respect to the Lebesgue measure on R, which writes p ∞ (x) = 2 σ 2 B(F ∞ (x)). We can now recall the description of the set of stationary distributions of the nonlinear process, which follows from [9, Proposition 4.1]. Proposition 3. Under Assumption (E), the stationary probability distributions for the nonlinear process (X t ) t≥0 are the translations of the probability distribution P ∞ ; that is to say, the probability distributions with cumulative distribution function x → F ∞ (x + x) for some x ∈ R.

Ergodicity results for the nonlinear diffusion process were obtained in [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF][START_REF] Jourdain | Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation[END_REF]. Let us precise that in [START_REF] Jourdain | Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation[END_REF], the stationary distributions are proven to be the translations of the function Ψ defined on (0, 1) by

∀u ∈ (0, 1), Ψ(u) := u v= 1 2 σ 2 2B(v)
dv.

Since Φ and Ψ have the same derivative, it is clear that the set of translations of Φ -1 coincides with the set of translations of Ψ -1 . As a consequence of Proposition 3, a stationary distribution for the nonlinear process is characterised by its expectation. In particular, P ∞ is the unique centered stationary distribution of the nonlinear process.

1.3. Main results and outline. We are now ready to state our main results and detail the outline of the paper.

Let us first recall the definition of the notion of chaoticity [11, Definition 2.1, p. 177]. If P n is a probability distribution on R n and k ∈ {1, . . . , n}, we denote by P k,n the marginal distribution of the k first coordinates under P n . Definition 4. For all n ≥ 1, let P n be a symmetric probability distribution on R n , and let P be a probability distribution on R. The sequence (P n ) n≥1 is said to be P -chaotic if, for all k ≥ 1, P k,n converges weakly to the product measure P ⊗k .

Recall that we denote by P n

∞ the unique stationary distribution of the projected particle system; it is the probability distribution on R n with density p n ∞ (z) with respect to the surface measure dz on M n . On the other hand, P ∞ refers to the unique centered stationary distribution of the nonlinear diffusion process; it is the probability distribution with density p ∞ with respect to the Lebesgue measure on R. Of course, our purpose is to establish the P ∞ -chaoticity of the sequence (P n ∞ ) n≥1 . Our proof is based on the study of the Laplace transform L exp(rΦ(u))du.

Besides, under Assumption (E), the point (4) in Lemma 2 ensures that as soon as r is taken in the set V := {r ∈ R : -2b(0)/σ 2 < r < -2b(1)/σ 2 }, then L ∞ (r) < +∞. This is the first part of Theorem 5.

Theorem 5. Under Assumption (E), (i) for all r ∈ V, L ∞ (r) is finite and writes

L ∞ (r) = 1 u=0
exp(rΦ(u))du;

(ii) for all (s, t) taken in the set

V 2 := {(s, t) ∈ V × V : s + t ∈ V},
there exists n 0 ≥ 2 such that, for all n ≥ n 0 , L 2,n ∞ (s, t) is finite and writes

L 2,n ∞ (s, t) = 1 n(n -1) n i=1 j =i J n i,j (s, t),
where, for all 1 ≤ i < j ≤ n,

J n i,j (s, t) := i-1 k=1 1 1 -(s + t) σ 2 2n k/n B(k/n) j-1 k=i 1 1 -t σ 2 2n k/n B(k/n) + s σ 2 2n 1-k/n B(k/n) n-1 k=j 1 1 + (s + t) σ 2 2n 1-k/n B(k/n) , while, for all 1 ≤ j < i ≤ n, J n i,j (s, t) := j-1 k=1 1 1 -(s + t) σ 2 2n k/n B(k/n) i-1 k=j 1 1 -s σ 2 2n k/n B(k/n) + t σ 2 2n 1-k/n B(k/n) n-1 k=i 1 1 + (s + t) σ 2 2n 1-k/n B(k/n) ; (iii) for all (s, t) ∈ V 2 , lim n→+∞ L 2,n ∞ (s, t) = L ∞ (s)L ∞ (t).
The point (ii) is proved in Section 2, while the point (iii) is detailed in Section 3.

As is stated in Corollary 8 below, Theorem 5 implies the P ∞ -chaoticity of P n ∞ , and actually yields the convergence of P k,n ∞ in a stronger sense than in Definition 4; namely, in Wasserstein distance [START_REF] Villani | Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften[END_REF]. Definition 6. Let k ≥ 1 and q ∈ [1, +∞). The Wasserstein distance of order q between two probability distributions µ and ν on R k is defined by

W q (µ, ν) := inf (X,Y )∈Π(µ,ν) (E[|X -Y | q ]) 1/q ,
where Π(µ, ν) refers to the set of pairs of random variables with marginal distributions µ and ν.

Remark 7. The definition of W q depends on the choice of the norm | • | on R k . But since all norms are equivalent on R k , all the associated distances W q are also equivalent. Therefore, convergence results for the W q topology do not depend on the choice of the underlying norm. In this paper, we take the convention that the Wasserstein distance of order q is defined with respect to the ℓ q norm |x| := (|x

1 | q + • • • + |x k | q ) 1/q on R k .
We derive chaoticity and convergence in Wasserstein distance as a corollary of Theorem 5.

Corollary 8. Under Assumption (E), (i) the sequence of stationary distributions P n ∞ of the projected particle system is P ∞ -chaotic, (ii) for all k ≥ 1, for all q ∈ [1, +∞),

lim n→+∞ W q P k,n ∞ , (P ∞ ) ⊗k = 0.
The proof of Corollary 8 is postponed to Appendix A. A summary of the long time and large scale behaviour of the projected particle system is detailed on Figure 1.

Projected particle system

Stationary distribution P n ∞ Nonlinear diffusion process Stationary distribution P ∞ n → +∞: propagation of chaos [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF] n → +∞: chaoticity (Corollary 8) t → +∞: ergodicity [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF] t → +∞: ergodicity [START_REF] Jourdain | Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation[END_REF] Figure 1. A summary of convergence results, in long time as well as for a large number of particles, for the projected particle system.

Figure 1 illustrates the fact that, when it makes sense, the interversion of the limits 'n → +∞' and 't → +∞' is generically correct for functionals of systems of rank-based interacting particles. This remark is of interest in the study of rank-based models of equity markets, such as the Atlas model introduced by Fernholz in the framework of Stochastic Portfolio Theory [START_REF] Fernholz | Stochastic portfolio theory[END_REF][START_REF] Banner | Atlas models of equity markets[END_REF][START_REF] Fernholz | Stochastic portfolio theory: A survey[END_REF]. Indeed, in this context, relevant quantities such as capital distribution curves or growth rates of portfolios are expressed in terms of the stationary distribution P n ∞ described in Proposition 1. The asymptotic behaviour of these quantities, when the size of the market grows to infinity, where investigated in [START_REF] Banner | Atlas models of equity markets[END_REF][START_REF] Chatterjee | A phase transition behavior for Brownian motions interacting through their ranks[END_REF]. On the other hand, it was suggested in [START_REF] Jourdain | Capital distribution and portfolio performance for rank-based models of equity market[END_REF] to use the propagation of chaos results of [START_REF] Jourdain | Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation[END_REF] to obtain a functional description of an infinite market first, and then apply the available ergodicity results on the nonlinear diffusion process to derive closed formulas for these relevant quantities. Corollary 8 is a first step toward the validation of the equivalence of both approaches, and we refer to [START_REF] Jourdain | Capital distribution and portfolio performance for rank-based models of equity market[END_REF] for a detailed account.

1.4. Notations. Throughout the paper, we use the following notations: for all s, t ∈ R, s ∧ t := min{s, t}, s ∨ t := max{s, t}, [s] -:= 0 ∨ (-s) and [s] + := 0 ∨ s. Besides, ⌊s⌋ denotes the integer part of s.

Expression of the Laplace transforms

This section is dedicated to the proof the point (ii) of Theorem 5. We first collect preliminary estimates in Subsection 2.1, and then compute the Laplace transform

L 2,n ∞ (s, t) of P 2,n ∞ in Subsection 2.2.
2.1. Preliminary estimates. Under Assumption (E), for all r ∈ V, for all n ≥ 1, for all k ∈ {1, . . . , n -1}, let us define

f + k,n (r) := r σ 2 2n k/n B(k/n) , f - k,n (r) := -r σ 2 2n 1 -k/n B(k/n) ,
so that the quantities J n i,j (s, t) introduced in Theorem 5 rewrite

J n i,j (s, t) := i-1 k=1 1 1 -f + k,n (s + t) j-1 k=i 1 1 -f + k,n (t) -f - k,n (s) n-1 k=j 1 1 -f - k,n (s + t) if i < j, J n i,j (s, t) := j-1 k=1 1 1 -f + k,n (s + t) i-1 k=j 1 1 -f + k,n (s) -f - k,n (t) n-1 k=i 1 1 -f - k,n (s + t) if i > j.
In this subsection, we exhibit upper bounds on the quantities f + k,n (r) and f - k,n (r), for r ∈ {s, t, s + t}, which ensure that the quantities J n i,j (s, t) are well defined for n large enough. We roughly proceed as follows: when k/n is far from 1, then k/n B(k/n) remains bounded by above, so that f + k,n (r) is arbitrarily small for n large enough. On the contrary, when k/n is close to

1, then k/n B(k/n) ≃ - 1 b(1)(1 -k/n) , so that f + k,n (r) ≃ -r σ 2 2b(1) 1 n -k ,
and the fact that r ∈ V provides natural bounds on the right-hand side. The same ideas allow to obtain similar bounds on f - k,n (r). We now give a rigorous formulation of these arguments. Under Assumption (E), for all ǫ > 0 such that ǫ < b(0) ∧ (-b(1)), we introduce

V ǫ := {r ∈ R : -2(b(0) -ǫ)/σ 2 < r < 2(-b(1) -ǫ)/σ 2 }, V ǫ 2 := {(s, t) ∈ V ǫ × V ǫ : s + t ∈ V ǫ }. For all r ∈ V ǫ , we define α + (r) := [r] + σ 2 2(-b(1) -ǫ) ∈ [0, 1), α -(r) := [r] -σ 2 2(b(0) -ǫ) ∈ [0, 1).
Let us now fix (s, t) ∈ V ǫ 2 . Then, for δ ∈ (0, 1/2) small enough,

• for all u ∈ [0, δ], B(u) ≥ u(b(0) -ǫ), • for all u ∈ [1 -δ, 1], B(u) ≥ (1 -u)(-b(1) -ǫ).
Besides, by Assumption (E), we have

m -(δ) := inf u∈[0,1-δ] B(u) u > 0, m + (δ) := inf u∈[δ,1] B(u) 1 -u > 0.
The heuristic arguments detailed at the beginning of the subsection translate into the following precise estimates: for all r ∈ {s, t, s + t}, for all n ≥ 1, for all k ∈ {1, . . . , n -1},

(6) f + k,n (r) = r σ 2 2n k/n B(k/n) ≤        [r] + σ 2 2nm -(δ) if k < n(1 -δ), [r] + σ 2 2(n -k) k/n -b(1) -ǫ ≤ α + (r) n -k if k ≥ n(1 -δ).
Similarly,

(7) f - k,n (r) = -r σ 2 2n 1 -k/n B(k/n) ≤        [r] -σ 2 2nm + (δ) if k > nδ, [r] -σ 2 2k 1 -k/n b(0) -ǫ ≤ α -(r) k if k ≤ nδ.
In particular, if n is chosen so that

[s + t] + σ 2 2nm -(δ) ≤ 1 2 ,
then we deduce from (6) that, for all k ∈ {1, . . . , n -1},

f + k,n (s + t) ≤ 1/2 if k < n(1 -δ), α + (s + t) if k ≥ n(1 -δ).
Similarly, if n is chosen so that

[t] + σ 2 2nm -(δ) + α -(s) ≤ α -(s) + 1 2 , [t] + σ 2 2nm -(δ) + [s] -σ 2 2nm + (δ) ≤ 1 2 , α + (t) + [s] -σ 2 2nm + (δ) ≤ α + (t) + 1 2 ,
then we deduce from ( 6) and ( 7) that, for all k ∈ {1, . . . , n -1},

f + k,n (t) + f - k,n (s) ≤      (α -(s) + 1)/2 if k ≤ nδ, 1/2 if nδ < k < n(1 -δ), (α + (t) + 1)/2 if k ≥ n(1 -δ).
These results are gathered together in the following lemma. Lemma 9. Let (s, t) ∈ V 2 . Under Assumption (E), there exists ǫ > 0 such that (s, t) ∈ V ǫ 2 . Let δ ∈ (0, 1/2) satisfying the conditions above. Let us define ᾱ ∈ (0, 1) by

ᾱ := max 1 2 , α + (s + t), α -(s + t), α + (s) + 1 2 , α -(s) + 1 2 , α + (t) + 1 2 , α -(t) + 1 2 .
Then, there exists n 0 ≥ 2 such that, for all n ≥ n 0 , for all k ∈ {1, . . . , n -1}, the quantities

1 -f + k,n (s + t), 1 -f + k,n (t) -f - k,n (s), 1 -f + k,n (s) -f - k,n (t), 1 -f - k,n (s + t) are larger than 1 -ᾱ. 2.2. Computation of L 2,n ∞ (s, t).
Let us first note that, since p n ∞ (z)dz is a symmetric probability distribution on R n , then for all symmetric and nonnegative function f : R n → R,

z∈Mn f (z 1 , . . . , z n )p n ∞ (z 1 , . . . , z n )dz = z∈Mn f (z 1 , . . . , z n )p n ∞ (z 1 , . . . , z n )dz,
where, for all z = (z 1 , . . . ,

z n ) ∈ M n , pn ∞ (z 1 , . . . , z n ) = n!1 {z1≤•••≤zn} 1 Z n exp 2 σ 2 n k=1 b n (k)z k .
Using the symmetry of p n ∞ (z)dz again, we deduce that, for all (s, t)

∈ V 2 , L 2,n ∞ (s, t) = 1 n(n -1) n i=1 j =i z∈Mn exp(sz i + tz j )p n ∞ (z)dz.
Let us now fix i ∈ {1, . . . , n} and j = i, and define

J n i,j (s, t) := z∈Mn exp(sz i + tz j )p n ∞ (z)dz.
Note that, at this stage, nothing prevents J n i,j (s, t) from being infinite. Using the parametrisation of M n by the n -1 coordinates x 1 = z 1 , . . . , x i-1 = z i-1 , x i+1 = z i+1 , . . . , x n = z n , so that the surface measure dz on M n rewrites

√ ndx 1 • • • dx i-1 dx i+1 • • • dx n , we obtain J n i,j (s, t) = n! √ n Z n (x1,...,xi-1,xi+1,...,xn)∈R n-1 1 {x1≤•••≤xi-1≤-(x1+•••+xi-1+xi+1+•••+xn)≤xi+1≤•••≤xn} × exp   tx j + k =i 2 σ 2 (b n (k) -b n (i)) -s x k   dx 1 • • • dx i-1 dx i+1 • • • dx n .
We denote

S := x 1 + • • • + x i-1 + x i+1 + • • • + x n ,
and let y k := x k + S, for all k = i, in the right-hand side above. Then we have

1 {x1≤•••≤xi-1≤-(x1+•••+xi-1+xi+1+•••+xn)≤xi+1≤•••≤xn} = 1 {y1≤•••≤yi-1≤0≤yi+1≤•••≤yn} , and
tx j + k =i 2 σ 2 (b n (k) -b n (i)) -s x k = ty j + k =i 2 σ 2 (b n (k) -b n (i)) -s y k -S   t + k =i 2 σ 2 (b n (k) -b n (i)) -s   . Note that S = k =i (y k -S), which implies S = 1 n k =i y k . Besides, k =i 2 σ 2 (b n (k) -b n (i)) -s = k =i 2 σ 2 b n (k) -(n -1) 2 σ 2 b n (i) -(n -1)s = -n 2 σ 2 b n (i) -(n -1)s,
tx j + k =i 2 σ 2 (b n (k) -b n (i)) -s x k = k =i γ k,n i,j (s, t)y k , where γ k,n i,j (s, t) := - s + t n + 2 σ 2 b n (k) + 1 {k=j} t. As a conclusion, J n i,j (s, t) = n! √ n nZ n (y1,...,yi-1,yi+1,...,yn)∈R n-1 1 {y1≤•••≤yi-1≤0≤yi+1≤•••≤yn} × exp   k =i γ k,n i,j (s, t)y k   dy 1 • • • dy i-1 dy i+1 • • • dy n = n! √ n nZ n J -,n i,j (s, t)J +,n i,j (s, t), where
J -,n i,j (s, t) := 0 yi-1=-∞ yi-1 yi-2=-∞ • • • y2 y1=-∞ exp(γ 1,n i,j (s, t)y 1 + • • • + γ i-1,n i,j (s, t)y i-1 )dy 1 • • • dy i-1 , J +,n i,j (s, t) := +∞ yi+1=0 +∞ yi+2=yi+1 • • • +∞ yn=yn-1 exp(γ n,n i,j (s, t)y n + • • • + γ i+1,n i,j (s, t)y i+1 )dy n • • • dy i+1 .
Let n 0 ≥ 2 and ᾱ ∈ (0, 1) be given by Lemma 9. We deduce from the definition of γ k,n i,j (s, t) that, if n ≥ n 0 , then, for all i, j ∈ {1, . . . , n} such that i = j, for all k ∈ {1, . . . , i -1},

γ 1,n i,j (s, t) + • • • + γ k,n i,j (s, t) = -(s + t) k n + 2n σ 2 B k n = 2n σ 2 B k n (1 -f + k,n (s + t)) ≥ 2n σ 2 B k n (1 -ᾱ) > 0,
and similarly, for all k ∈ {i + 1, . . . , n},

γ k,n i,j (s, t) + • • • + γ n,n i,j (s, t) = -(s + t) n -k + 1 n + t - 2n σ 2 B k -1 n = - 2n σ 2 B k -1 n (1 -f + k-1,n (t) -f - k-1,n (s)) ≤ - 2n σ 2 B k -1 n (1 -ᾱ) < 0,
which ensures that J -,n i,j (s, t) and J +,n i,j (s, t) are finite and write

J -,n i,j (s, t) = i-1 k=1 1 γ 1,n i,j (s, t) + • • • + γ k,n i,j (s, t) , J +,n i,j (s, t) = n k=i+1 -1 γ n,n i,j (s, t) + • • • + γ k,n i,j (s, t)
, thanks to successive integrations. This finally gives

J n i,j (s, t) = n! √ n nZ n i-1 k=1 1 -(s+t)k n + 2n σ 2 B( k n ) j-1 k=i 1 -tk+s(n-k) n + 2n σ 2 B( k n ) n-1 k=j 1 (s+t)(n-k) n + 2n σ 2 B( k n ) if i < j, and
J n i,j (s, t) = n! √ n nZ n j-1 k=1 1 -(s+t)k n + 2n σ 2 B( k n ) i-1 k=j 1 -sk+t(n-k) n + 2n σ 2 B( k n ) n-1 k=i 1 (s+t)(n-k) n + 2n σ 2 B( k n ) if i > j.
To complete the proof, we remark that

L 2,n ∞ (0, 0) = 1 = n! √ n nZ n n-1 k=1 1 2n σ 2 B( k n )
, which allows us to get rid of the constant term n! √ n nZn and to obtain the expected expression of J n i,j (s, t) in the point (ii) of Theorem 5, for (s, t) ∈ V 2 .

Convergence of the Laplace transforms

The proof of the point (iii) of Theorem 5 works in two steps: first, we prove that, for all t ∈ V, the Laplace transform

L 1,n ∞ (t) = L 2,n ∞ (t, 0) of P 1,n ∞ converges to L ∞ (t)
. Second, we check that, for (s, t) ∈ V 2 , the difference between L 2,n ∞ (s, t) and the product L 1,n ∞ (s)L 1,n ∞ (t) vanishes. These two steps are addressed in the respective Subsections 3.2 and 3.3. The preliminary Subsection 3.1 gathers useful elementary results.

3.1. Elementary inequalities. We shall use the following inequalities, which are elementary consequences of the Taylor-Lagrange inequality. (TL1) For all α ∈ [0, 1), for all x ∈ [-α, +∞),

| log(1 + x) -x| ≤ κ(α)x 2 , where κ(α) := 1 2(1-α) 2 . (TL2) For all x, y ∈ R, | exp(x) -exp(y)| ≤ exp(y) (|x -y| + |R(x -y)|), where the function R : z → exp(z) -1 -z is such that, for all C ∈ [0, +∞), for all z ∈ [-C, C], |R(z)| ≤ 1 2 exp(C)z 2 .
In particular, we deduce from (TL1) that, for all C ∈ [0, +∞), ( 8)

lim n→+∞ 1 - C n -n = exp(C). 3.2. Convergence of L 1,n ∞ (t). Let us fix t ∈ V.
By the results of Section 2, there exists n 0 ≥ 2 such that, for all n ≥ n 0 , the Laplace transform

L 1,n ∞ (t) = L 2,n ∞ (t, 0) of p 1,n ∞ is finite and writes L 1,n ∞ (t) = 1 n n i=1 I n i (t), I n i (t) := i-1 k=1 1 1 -t σ 2 2n k/n B(k/n) n-1 k=i 1 1 + t σ 2 2n 1-k/n B(k/n) .
Then, we have

(9) |L 1,n ∞ (t) -L ∞ (t)| ≤ n i=1 i n u= i-1 n |I n i (t) -exp(tΦ(u))|du.
Let ǫ > 0 and δ ∈ (0, 1/2) be given by Subsection 2.1 for the pair (t, 0) ∈ V 2 . We split the sum appearing in the right-hand side of (9) into boundary terms, corresponding to i ≤ nδ and i ≥ n(1 -δ), and a central term, corresponding to nδ < i < n(1 -δ). These terms are addressed separately, in the respective §3.2.1 and §3.2.2.

Boundary terms. For all

n ≥ n 0 , i≤nδ i n u= i-1 n |I n i (t) -exp(tΦ(u))|du ≤ 1 n ⌊nδ⌋ i=1 I n i (t) + δ u=0 exp(tΦ(u))du.
It is an easy consequence of the point (i) in Theorem 5 that the integral in the right-hand side above vanishes with δ. The purpose of this paragraph is to show that [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF]

lim δ↓0 lim sup n→+∞ 1 n ⌊nδ⌋ i=1 I n i (t) = 0.
Let us first assume that t ≥ 0. Then, for all i ≤ nδ,

n-1 k=i 1 1 + t σ 2 2n 1-k/n B(k/n) ≤ 1.
We now use the fact that, if i ≤ nδ, then for all k ∈ {1, . . .

, i -1}, k/n B(k/n) ≤ 1 b(0)-ǫ , to write i-1 k=1 1 1 -t σ 2 2n k/n B(k/n) ≤ 1 - tσ 2 2n(b(0) -ǫ) -(i-1) ≤ 1 - tσ 2 2n(b(0) -ǫ) -n
, as soon as n is large enough to ensure that tσ 2 /(2n(b(0) -ǫ) < 1. Using (8), we deduce that

lim sup n→+∞ 1 n ⌊nδ⌋ i=1 I n i (t) ≤ δ exp tσ 2 2(b(0) -ǫ) ,
and (10) easily follows.

Let us now assume that t < 0. Then, for i ≤ nδ, we still have the rough bound

i-1 k=1 1 1 -t σ 2 2n k/n B(k/n) ≤ 1,
but we need to be more careful as far as the product

n-1 i=k 1 1 + t σ 2 2n 1-k/n B(k/n) = n-1 i=k 1 1 -f - k,n (t)
is concerned. On the one hand,

log   n-1 k=⌊nδ⌋+1 1 1 -f - k,n (t)
  = - n-1 k=⌊nδ⌋+1 log 1 -f - k,n (t) ,
and combining Lemma 9 with the inequality (TL1) yields

- n-1 k=⌊nδ⌋+1 log 1 -f - k,n (t) ≤ n-1 k=⌊nδ⌋+1 f - k,n (t) + κ(ᾱ) f - k,n (t) 2 .
Since the definition of

f - k,n (t) yields n-1 k=⌊nδ⌋+1 f - k,n (t) = - tσ 2 2n n-1 k=1 1 {k/n>δ} 1 -k/n B(k/n) while (7) implies n-1 k=⌊nδ⌋+1 f - k,n (t) 2 ≤ 1 n tσ 2 2m + (δ) 2 , we deduce that lim n→+∞ log   n-1 k=⌊nδ⌋+1 1 1 -f - k,n (t)   = - tσ 2 2 1 v=δ 1 -v B(v) dv.
On the other hand, [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF] gives

⌊nδ⌋ k=i 1 1 -f - k,n (t) ≤ ⌊nδ⌋ k=i 1 1 -α/k ,
where α := α -(t) < 1. Using (TL1) again, we write

log   ⌊nδ⌋ k=i 1 1 -α/k   = - ⌊nδ⌋ k=i log 1 - α k ≤ ⌊nδ⌋ k=i α k + κ(α) α 2 k 2 ≤ α ⌊nδ⌋ k=i 1 k + κ(α)α 2 π 2 6 ≤ α (1 + log(nδ) -log(i)) + κ(α)α 2 π 2 6 , so that ⌊nδ⌋ k=i 1 1 -f - k,n (t) ≤ K(α)δ α 1 (i/n) α ,
where K(α) := exp(α + κ(α)α 2 π 2 /6). Since

lim n→+∞ 1 n ⌊nδ⌋ i=1 1 (i/n) α = δ v=0 dv v α = δ 1-α 1 -α , we conclude that lim sup n→+∞ 1 n ⌊nδ⌋ i=1 I n i (t) ≤ K(α) 1 -α δ exp - tσ 2 2 1 v=δ 1 -v B(v) dv =: M (δ).
To obtain [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF], we now have to check that M (δ) vanishes with δ. To this aim, we fix 0 < η < b(0) ∧ (-b(1)) such that t ∈ V η . Since the diverging integral

1 v=δ 1-v B(v)
dv is equivalent to -log(δ)/b(0) when δ vanishes, we deduce that, for δ small enough, we have

1 v=δ 1 -v B(v) dv ≤ -log δ b(0) -η , so that exp - tσ 2 2 1 v=δ 1 -v B(v) dv ≤ δ -β , with β := -tσ 2 2(b(0) -η) ∈ (0, 1).
As a conclusion, M (δ) is of order δ 1-β when δ is small, whence [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF]. The boundary term corresponding to i ≥ n(1 -δ) can be handled by symmetric arguments.

3.2.2. Central term. We now prove that [START_REF] Sznitman | Topics in propagation of chaos[END_REF]

lim δ↓0 lim sup n→+∞ ⌊n(1-δ)⌋-1 i=⌊nδ⌋+1 i n u= i-1 n |I n i (t) -exp(tΦ(u))|du = 0.
To this aim we fix i ∈ {⌊nδ⌋ + 1, . . . , ⌊n(1 -δ)⌋ -1} and u ∈

[ i-1 n , i n ]. By (TL2), |I n i (t) -exp(tΦ(u))| ≤ exp(tΦ(u))(|∆| + |R(∆)|), where ∆ := ∆ 1 + ∆ 2 , with ∆ 1 := - i-1 k=1 log 1 - tσ 2 2n k/n B(k/n) - tσ 2 2 u v=0 v B(v) dv, ∆ 2 := - n-1 k=i log 1 + tσ 2 2n 1 -k/n B(k/n) + tσ 2 2 u v=0 1 -v B(v) dv.
For all k ∈ {1, . . . , i -1}, we deduce from Lemma 9, the inequality (TL1) and the estimate ( 6) that

|∆ 1 | ≤ i-1 k=1 tσ 2 2n k/n B(k/n) - tσ 2 2 u v=0 v B(v) dv + κ(ᾱ) n tσ 2 2m -(δ) 2 ≤ |t|σ 2 2 i-1 k=1 k n v= k-1 n k/n B(k/n) - v B(v) dv + |t|σ 2 2 u v= i-1 n v B(v) dv + κ(ᾱ) n tσ 2 2m -(δ) 2 .
Using the uniform continuity of v/B(v) on [0, 1 -δ], we deduce that for n large enough, for all k ∈ {1, . . . , ⌊n(1 -δ)⌋ -1},

∀v ∈ k -1 n , k n , k/n B(k/n) - v B(v) ≤ δ.
As a consequence,

|∆ 1 | ≤ |t|σ 2 2 δ + 1 n |t|σ 2 2m -(δ) + κ(ᾱ) tσ 2 2m -(δ) 2 =: M 1 (n, δ),
and we note that M 1 (n, δ) does not depend on i ∈ {⌊nδ⌋ + 1, . . . , ⌊n(1 -δ)⌋ -1} and satisfies

lim δ↓0 lim sup n→+∞ M 1 (n, δ) = 0.
We similarly construct M 2 (n, δ) satisfying the same conditions as M 1 (n, δ) and such that |∆ 2 | ≤ M 2 (n, δ) for n large enough. As a consequence, for δ > 0 small enough and n large enough, we have

|∆ 1 | + |∆ 2 | ≤ 1, so that (TL2) yields |I n i (t) -exp(tΦ(u))| ≤ exp(tΦ(u))(M 1 (n, δ) + M 2 (n, δ) + exp(1) 2 (M 1 (n, δ) + M 2 (n, δ)) 2 ),
and finally

⌊n(1-δ)⌋-1 i=⌊nδ⌋+1 i n u= i-1 n |I n i (t) -exp(tΦ(u))|du ≤ M 1 (n, δ) + M 2 (n, δ) + exp(1) 2 (M 1 (n, δ) + M 2 (n, δ)) 2 1 u=0 exp(tΦ(u))du,
which completes the proof of (11).

3.3. Convergence of L 2,n ∞ (s, t) -L 1,n ∞ (s)L 1,n ∞ (t).
Let n 0 ≥ 2 be given by Lemma 9. Then for all n ≥ n 0 ,

|L 2,n ∞ (s, t) -L 1,n ∞ (s)L 1,n ∞ (t)| = 1 n(n -1) n i=1 j =i J n i,j (s, t) - 1 n 2 n i=1 n j=1 I n i (s)I n j (t) ≤ 1 n(n -1) n i=1 j =i |J n i,j (s, t) -I n i (s)I n j (t)| + 1 n(n -1) n i=1 I n i (s)I n i (t) + 1 n -1 L 1,n ∞ (s)L 1,n ∞ (t).
By the results of Subsection 3.2, the last term in the right-hand side above vanishes when n grows to infinity. The diagonal term

I n i (s)I n i (t) = 0.
To this aim, we write, for all i ∈ {1, . . . , n},

I n i (s)I n i (t) = i-1 k=1 1 1 -sσ 2 2n k/n B(k/n) 1 1 -tσ 2 2n k/n B(k/n) n-1 k=i 1 1 + sσ 2 2n 1-k/n B(k/n) 1 1 + tσ 2 2n 1-k/n B(k/n)
, and note that if st ≥ 0, then I n i (s)I n i (t) ≤ I n i (s + t), so that (12) follows from the results of Subsection 3.2. On the other hand, if st < 0, say s < 0 < t, then

I n i (s)I n i (t) ≤ i-1 k=1 1 1 -tσ 2 2n k/n B(k/n) n-1 k=i 1 1 + sσ 2 2n 1-k/n B(k/n) .
Let us fix ǫ > 0 and δ ∈ (0, 1/2) as in Subsection 2.1. Arguing as in §3.2.1, we obtain

lim sup n→+∞ 1 n ⌊nδ⌋ i=1 i-1 k=1 1 1 -tσ 2 2n k/n B(k/n) n-1 k=i 1 1 + sσ 2 2n 1-k/n B(k/n) ≤ exp tσ 2 2(b(0) -ǫ) M (δ) < +∞,
and the same arguments apply to the sum for i ≥ n(1 -δ). On the other hand, combining the estimates ( 6), ( 7) with ( 8) yields

lim sup n→+∞ ⌊n(1-δ)⌋-1 i=⌊nδ⌋+1 i-1 k=1 1 1 -tσ 2 2n k/n B(k/n) n-1 k=i 1 1 + sσ 2 2n 1-k/n B(k/n) ≤ exp tσ 2 2m -(δ) - sσ 2 2m + (δ)
.

We deduce that

lim sup n→+∞ 1 n n i=1 I n i (s)I n i (t) < +∞, whence (12)
(1 -f ± k,n (s) -f ± k,n (t)) -log(1 -f ± k,n (s)) -log(1 -f ± k,n (t))
, where Lemma 9 ensures that each term f ± k,n (s), f ± k,n (t) and f ± k,n (s) + f ± k,n (t) is lower than ᾱ < 1. As a consequence, (TL1) yields We deduce from the estimates ( 6) and ( 7) that To complete the proof of (13), we now check that I n i (s)I n j (t)

| log(1 -f ± k,n (s) -f ± k,n (t)) -log(1 -f ± k,n (s)) -log(1 -f ± k,n (t))| ≤ κ(ᾱ)((f ± k,n (s) + f ± k,n (t)) 2 + (f ± k,n (s)) 2 + (f ± k,n (t)) 2 ) ≤ 3κ(ᾱ)((f ± k,n (
F n i (s) ≤ 1 n sσ 2 2m -(δ)
M ′ (δ) n = M ′ (δ) n L 1,n ∞ (s)L 1,n ∞ (t)

1 . Introduction 1 . 1 .

 111 Rank-based interacting diffusions. Let b : [0, 1] → R be a continuous function, and σ = 0.

where (W 1

 1 (t), . . . , W n (t)) t≥0 is a standard Brownian motion in R n , and for all k ∈ {1, . . . , n}, b n (k) := n

  n (k) = n b = 0 by Assumption (E). As a consequence,

  I n i (s)I n j (t) {• • • } = 0,where {• • • } refers to the braced term in the right-hand side of (14). Note that, on account of the results of §3.3.1, it is equivalent to show that lim

vanishes when n

  grows to infinity. On the other hand, (s)I n j (t)C ′ 1 {i≤nδ} = C ′ L ∞ (t) lim sup n→+∞ 1 n ⌊nδ⌋ i=1 I n i (s),and it was proved in §3.2.1 that the last term in the right-hand side vanishes with δ. Addressing the other boundary terms similarly, we obtain (13) and thereby complete the proof of the point (iii) in Theorem 5.

  ∞ . Following the results of Subsection 1.2, we can already obtain an explicit expression of L ∞ (r); indeed, since the inverse of the cumulative distribution function F ∞ of P ∞ is Φ, then, for all r ∈ R,

	of P L ∞ (r) =	1 u=0
	2,n ∞ (s, t) :=	z∈Mn	exp(sz 1 + tz 2 )p n ∞ (z)dz
	of P 2,n ∞ , and of the Laplace transform
	L ∞ (r) :=	x∈R	exp(rx)p ∞ (x)dx

  (s, t) writes as a sum, for k ∈ {1, . . . , n -1}, of terms of the form log

	.
	3.3.2. Main term. In this paragraph, we finally check that
	(13)	lim n→+∞	1 n(n -1)	n i=1 j =i	|J n i,j (s, t) -I n i (s)I n j (t)| = 0.
	|J n i,j (s, t) -I n i (s)I n j (t)| ≤ I n i (s)I n j (t) |ρ n i,j (s, t)| + |R(ρ n i,j (s, t))| ,
	where
	Remark that ρ n i,j	ρ n i,j (s, t) := log(J n i,j (s, t)) -log(I n i (s)I n j (t)).

By (TL2), we have, for all i, j ∈ {1, . . . , n} such that i = j,

  so that that there exist a nonnegative and finite constant M (δ), that depends on δ, and a nonnegative and finite constant C, that does not depend on δ, such that {i≤nδ}+ 1 {i≥n(1-δ)} + 1 {j≤nδ} + 1 {j≥n(1-δ)} .For n large enough, the right-hand side above is lower that 3C, so that (TL2)yields {i≤nδ} + 1 {i≥n(1-δ)} + 1 {j≤nδ} + 1 {j≥n(1-δ)} + C 2 1 {i≤nδ} + 1 {i≥n(1-δ)} 2 + C 2 1 {j≤nδ} + 1 {j≥n(1-δ)} + C 2 1 {i≤nδ} + 1 {i≥n(1-δ)} + 1 {j≤nδ} + 1 {j≥n(1-δ)} .As a consequence, there exist a nonnegative and finite constant M ′ (δ), that depends on δ, and a nonnegative and finite constant C ′ , that does not depend on δ, such that, for n large enough, for all i = j in {1, . . . , n},

					2	2	+	2m + (δ) 2 sσ 2	2
			+	π 2 6	|s|σ 2 2(b(0) -ǫ)	2	1 {i≤nδ} +	|s|σ 2 2(-b(1) -ǫ)	2	1 {i≥n(1-δ)} ,
	|ρ n i,j (s, t)| ≤ + C 1 |R(ρ n M (δ) n i,j (s, t))| ≤ exp(3C) 2 M (δ) n + C 1 2
	≤	3 2	exp(3C)	M (δ) 2 n 2 2
	= n 2 (14) 3 2 M (δ) 2 exp(3C) |J n i (s)I n j (t)| i,j (s, t) -I n ≤ I n i (s)I n j (t) M ′ (δ) n	+ C

′ 1 {i≤nδ} + 1 {i≥n(1-δ)} + 1 {j≤nδ} + 1 {j≥n(1-δ)} .

Acknowledgements. The author is grateful to Benjamin Jourdain for his very helpful comments on this work.

Appendix A. Proof of Corollary 8

Proof of Corollary 8. Since V 2 is an open subset of R 2 containing (0, 0), the point (iii) of Theorem 5 implies the weak convergence of P 2,n ∞ to the product measure (P ∞ ) ⊗2 . According to the proof of [11, Proposition 2.2, p. 177], this is enough to ensure the P ∞ -chaoticity of the sequence (P n ∞ ) n≥1 , which is the point (i) of Corollary 8.

Let us now address the point (ii) and fix k ≥ 1, q ∈ [1, +∞). Following Remark 7 and [13, Theorem 6.9], to prove that P k,n ∞ converges to (P ∞ ) ⊗k in Wasserstein distance of order q, it suffices to check that

where, for all n ≥ 1, (X n 1 , . . . , X n n ) is distributed according to P k,n ∞ , while (X 1 , . . . , X n ) is distributed according to (P ∞ ) ⊗k . Using the linearity of the expectation and the symmetry of P n ∞ , we deduce that it is enough to check this result for k = 1. Then we already know that X n 1 converges in distribution to X 1 , and we now check that the sequence of random variables (|X n 1 | q ) n≥1 is uniformly integrable, which implies the convergence of E[|X n 1 | q ] and completes the proof. To check the uniform integrability of the sequence (|X n 1 | q ) n≥1 , we fix r > q and prove that the sequence (E[|X n 1 | r ]) n≥1 is bounded. To this aim, we fix ρ > 0 such that -ρ ∈ V and ρ ∈ V. Then, there exists M ≥ 0 such that, for all x ∈ R, |x| r ≤ M + (exp(-ρx) + exp(ρx)), so that, for all n ≥ 1,

), and we deduce from the point (iii) of Theorem 5 that the right-hand side converges to a finite value when n grows to infinity, which implies that the left-hand side is uniformly bounded with respect to n.