Marie-José Gremmo

Advising for Language Learning: Interactive Characteristics and Negotiation Procedures

The first principle which constitutes the specific character of self-directed language learning schemes (SDLLS) is that they enable learners to have direct access to learning resources: the focus is on the learner"s autonomous 1 learning activity. The second principle for SDLLS is that they offer learners direct access to a pedagogical device which allows them to become competent self-directed learners. Both principles are essential: on the one hand, giving learners full responsibility for the learning activity in terms of freedom of choice and power of control 2 allows them to experience self-directed learning to the best of their capacity; on the other hand, giving them the possibility to learn-to-learn allows them to further develop their mastery of the learning process and their own specific learning competence: learner autonomy then becomes real.

The specific pedagogical device, which deals with the second objective of helping the development of the learner"s learning competence and ability to self-direct is now generally termed "language advising" 3 . In this article, my aim is to show the essential part that verbal interaction plays in language advising. It is through the specific interactional characteristics of the "advising conversation" that language advisers are able to help learners without taking control or imposing decisions, thus making advising sessions a meaningful developmental resource for learners. I will first give a synthetic description of the nature of language 1 Autonomy, as defined by Holec (1979) is "the ability to take charge of one"s own learning", that is "to have the responsibility for all the decisions concerning all aspects of the learning: determining objectives, defining the contents and progressions, selecting methods and techniques to be used, monitoring the procedure of acquisition and evaluating what has been acquired". 2 Bouchard (2002) 3 This term is now widespread, although in some SDLLS other terms may be in use.

advising. Then I will illustrate it using extracts from five advising conversations that took place between a specific adviser/learner pair. Finally, I will draw more general conclusions which will discuss aspects of the language interaction for advising and some requisites for advising conversations.

The nature of language advising

Language advising is a pedagogical device based on human interaction: the interaction that learners have with a supporting "expert", the language adviser made available by the institution. Learners and advisers interact in a one-to-one relationship, and their interaction is focused on the learner"s activity. Advising for self-directed learning must be clearly differentiated from individualized teaching: although individualized teaching takes the learner"s specific characteristics into account and gives learners some latitude in the monitoring of their work, the pedagogical control remains in the hands of the teacher. In contrast, advising for self-directed learning is based on the fact that learners can and do take their own decisions, and make conscious and meaningful choices throughout the learning process. In other words, SDLLS are specific in that they are institutional schemes offering both learning a language "without being taught" (Holec, Little & Richterich, 1996), and learning to learn a language. As the aim of language advising is to ensure the development of the learner"s learning competence, advising is: a) focused on the learning process much more than on the learning content, b) non-decisional, and not founded on a power relationship, c) retro-active, negotiative , non-programmable and not programmed, d) dealing with a specific coherent conceptual framework, that of language didactics. One can see that the role of the adviser is far different from the role of the teacher, who selects the knowledge to be acquired, prepares the ways in which it is learnt, monitors the teaching/learning sessions and finally evaluates the results. The adviser"s role rests not on a pro-active adaptive monitoring of learning activities (as individualized teaching does) but on a retro-active contextualised monitoring, depending on the learner"s request, bringing focused help to the learner"s construction of his/her learning competence, and taking place both before and after the learner"s work sessions. The aim of this article is to clarify the characteristics of language advising, pointing out the way these characteristics define the modalities of the adviser"s action in an advising conversation. A case study will be used as a basis for the discussion.

The case study

The extracts relate to a specific adviser/learner pair which was part of a larger research study4 .

Both are women. Lorraine, the learner, is a 24-year-old fourth-year economics student and has registered for the SDLLS because she plans to go through a selection procedure for a postgraduate diploma which includes an English test. Nancy, the adviser, is a 40-year-old experienced adviser. The advising sessions took place in French, which was the mother tongue of them both5 . Two sets of data were collected. Firstly, the eleven advising sessions that took place between them were video-recorded. The extracts presented here deal with work on oral comprehension. Secondly, Nancy and Lorraine were interviewed separately, and asked to discuss their experience of the advising sessions. Their interviews were also recorded.

The adviser's verbal behaviour as a basis for advising

During Advising Session 1 (AS1), the SDLLS was presented to Lorraine. She discussed her learning objectives with her adviser Nancy, and together they set up a work programme with a first batch of resources. She took an active part in selecting the resources, accepting or rejecting Nancy"s offers ("OK, I will test that" or "No I don't think I'll use that one").

Advising Session 2 (AS2) began with Lorraine"s report of the work she had done. She had worked on two resources: a "classic" course book including oral comprehension activities which I will call BS, and a specific oral comprehension handbook, presenting working units 4 built around a near-authentic recording and exercises, which I will call RL. Lorraine stated at the very beginning:

1. I have tested everything (...) I like BS, it is rather simple, there are small exercises, it was not very hard and I managed well in the whole unit6

This evaluation did not draw remarks from Nancy and Lorraine went on: d) advising deals with a specific coherent conceptual framework, that of language didactics: quotations 4 and 5 clearly indicate that the information given by Nancy deals exclusively with concepts and tools relevant to the field of language didactics. Moreover, the information given is not presented as "the best way" to learn: it is much more focused on giving expert criteria to get Lorraine to re-assess her analysis and leave her the freedom and responsibility of improving "her own way".

At that point, one could argue that a more clearly "directive" decision-making from Nancy, as one would expect from a teacher, would be more effective. However, it is very important to realize that the nature of self-directed learning itself makes any directivity ineffective. As the adviser"s activity is clearly separate from the learner"s activity and the advising process is clearly separate from the learning process, advisers cannot indeed impose on learners, let alone enforce, any decision they would make "in the learner"s interest". Moreover, any decision by the adviser which gives priority to the acquisition of the linguistic content will slow down the development of the learner"s autonomy, which is essential for efficient selfdirected learning.

Advisers are thus conscious that there is a tension between the objective of linguistic acquisition and the objective of learning competence development, and that the "learning to learn" objective requires time. Nancy made this very explicit to Lorraine, as in AS1:

8. (…) it is important for you to take the time to test various materials, so that we will have a better idea of what you like

The verbal interaction as a trigger for behavioural change.

The following advising session (AS3) opened with this remark from Lorraine:

Well in fact there is a small problem and I would like us to change the materials, RL especially, because I have a lot of problems in terms of oral comprehension and perhaps it would be better if we chose to work on oral expression, so that I will speak and surely afterwards my comprehension will improve

Nancy then engaged in a discussion of Lorraine"s comprehension difficulties, seeming to take no notice of Lorraine"s decision to drop comprehension work altogether:

10. Could you describe some of your problems to me? What were you working on? (…) and how did you proceed?

This question was answered in the following manner by Lorraine;

11. I read the questions before now, I try to find what it is about and then I try to note down everything.that I have managed to understand and to answer the questions at the same time (…) then I stop [ the cassette ] and listen again, quite often for sure! Lorraine indicated thus that, contrary to what she had induced Nancy (and us) to believe from what she said in AS2, she had put Nancy"s advice into practice: she had indeed worked on RL again, and she had worked with the procedure that Nancy had described to her (cf. quotations 3 and 5).

During AS3, Lorraine was interactively quite active and the exchanges were more numerous than in AS2. As Lorraine did not fully agree with the procedure that Nancy had suggested, they rediscussed the information given by Nancy in AS2. Lorraine was asked to express the way in which she saw things:

12. there are many things which I do not understand and I feel that I am not getting anywhere and it irritates me

because of the fact that I could not understand immediately, I wasted time listening again and again

Nancy was then able to add more detailed information relevant to learning to learn, meanwhile giving Lorraine psychological support: The extracts here illustrate how advising can trigger off behavioral change in learners through the negotiation process which is at the heart of the advising conversation. In AS2, Nancy had enforced no decision on Lorraine, but she did give herself the right to discuss Lorraine"s positive account of her comprehension method (cf. quotations 2 and 3) in the light of her own didactic expertise. She then gave Lorraine information on the didactic concept of comprehension and suggested to her new methodological procedures for comprehension activities (quots. 4 and 5). In the same manner, in AS3, she allowed herself to discuss

14. I
Lorraine"s decision to change her linguistic objectives, meanwhile she explicitly acknowledged Lorraine"s total freedom of decision (quots. 14 and 15). Her advising behavior gave Lorraine the opportunity to re-assess her own approach. One important element that the extracts show is that the negotiation that takes place between adviser and learner is valuable because it enables learners to engage in an internal negotiation procedure with themselves.

When Lorraine started her self-directed work after AS2, she obviously re-negotiated with herself the decision expressed in AS2, using the information that Nancy had provided her during the advising session, and she decided on her own to try out her adviser"s methodological suggestions. Quotations 12 and 13 show that Lorraine was not fully satisfied with the experimentation since she felt irritated and found she was not very efficient, but in return she had gained an experience which allowed the advising conversation in AS3 to go beyond what was reached in AS2. Quotation 16 demonstrates how Lorraine kept on a personal negotiating process which brought her to finally internalize some of the criteria that her adviser had discussed.

This stresses another important dimension of advising. The adviser"s action trusts in the learner"s capacity to adapt the adviser"s contribution to his/her advantage. This is why, as we see in our case study, advisers can be effective without taking decisions in the learner"s place.

But on the other hand, they can only be effective if they make their "expert" criteria explicit to the learners. Their action is thus neither imposition nor "laissez-faire": it is based on reiterative negotiation.

Advising as a trigger for non-programmable change.

It is obvious from what I have discussed above that some learning to learn did occur in

Lorraine. But what her case study also shows is that there is a discrepancy between what the adviser hopes for in terms of learner change and the changes that really occur to the learner.

First of all, quotation 16 shows that advising is dependent on the learning context. In AS4, Lorraine briefly mentioned her comprehension work in passing: the issue of oral comprehension did not seem to be vital to her any more so it was no longer an important conversational topic. Even if Nancy had wanted to discuss it (which she obviously did not),

the simple fact that it had ceased to be a cognitive focus for Lorraine would have made the discussion useless. Lorraine"s freedom of choice and control goes as far as the areas where change may occur.

Secondly, a very interesting finding was uncovered through the comparison of the two sets of data7 : the data from the recordings of the advising session differed from both Nancy"s evaluation of Lorraine"s evolution and from Lorraine"s own description, as revealed in the interviews.

In her interview, Lorraine showed that she had a clear understanding of the advising sessions: and she was under the impression that her advising role was not understood by Lorraine:

21. I felt that Lorraine considered that my role was to provide her with materials and that all the rest did not really interest her

It seems to me that the difference between Lorraine"s and Nancy"s evaluations and the conclusions that can be drawn from the recordings of advising sessions can be explained by considering other dimensions that play a part in the advising conversation.

The analysis of the recorded sessions in fact confirms that the way in which Lorraine talked about her oral comprehension work evolved little: she kept analyzing the difficulties in terms of length, speech speed and vocabulary. It shows that her words reflect the abstract principles which underlie her conception of language learning. But her learning procedures for oral comprehension deeply changed: whereas in AS3 she said she was going to drop comprehension work altogether, not only did she continue working with comprehension resources but she also did so in more varied ways (using short and longer materials, didactic 10 or authentic materials, sound or video cassettes). In her interview, when the researcher asked her to comment on this point, she stated:

22. It is true that at the beginning these materials frightened me, the length, the speed, but it was perhaps just a question of confidence, and through constant repetition, I managed to understand better, and she clearly attributes this evolution to Nancy"s advice:

23. Nancy insisted, she encouraged me to try again while saying " if that is really too difficult we will stop" so I kept on with this method whereas I would perhaps have given it up and finally I have started to like it This is how she conceives of the adviser"s role:

the adviser is there to give you a push when you can't manage to work as you'd like to

If we put quotations 17, 22 and 23 together, we can again see the importance of the verbal interaction between learner and adviser.

The difference between Nancy"s rather negative appreciation of Lorraine"s development and the picture that emerges from the analysis of the advising session can be explained, in my view, because it reflects a difference between what can be expected from advising and what is actually achieved through the interaction of a specific learner and a specific adviser. In terms of general expectations, Nancy"s criteria for evaluation are mainly situated at the level of Lorraine"s decisional principles (quot. 21). But Lorraine"s evolution is on the level of her learning behavior, for instance, putting up with her lack of performance (such as not understanding all the words), or using new procedures (such as listening several times to the same extract).

Again, the data show that Lorraine"s behavioural change is founded on her understanding of the advising session, and more specifically of the nature of the verbal interaction (quot. 25), the interactive roles (quot. 26) and the expertise of the adviser (quot. 27). She also never felt either constrained or imposed upon by Nancy (quot. 17).  The influence that advisers have on the development of learners" learning autonomy is determined by the interaction modes which they set up during the advising sessions, and in particular by the verbal explanation of their advising role, which enables them to make the advising situation understandable to learners. Learners then feel able, as

at the beginning I thought that it was like an individual class then

Lorraine did, to take up new options (through trial and error) and to control their learning process (through acceptance or refusal of the adviser"s help);

 The evolution that learners undergo with the help of advising conversations is never exactly what advisers expect or work for, as learners" free will and understanding of the situation are the main trigger to change. Nevertheless, one can doubt if change would ever occur without the determined action of advising.

Language for advising

The case study presented here vividly illustrates how powerful an organizational tool language is for the development of learning autonomy. As I have already pointed out, the toand-fro movement between experiencing self-directed learning and reflecting on this experience is fundamental to the whole concept of SDLLS. In advising sessions, learners and advisers do not carry out any language learning, they "talk about" language learning. But they do not talk about language learning "in general". On the contrary, they talk about the "real" and contextualized learning activity that a specific learner is experiencing. Thus, for adequate advising to take place, there must be effective learning activity, whatever the type, the content or the quality8 . The advising session is thus used to set an interval of time during which learners can engage in the metacognitive reflection necessary for the development of their learning competence.

The advising session is also used as an interface between the learners and their activity: the act of talking to the adviser about what they have done, or what they plan to do, creates a symbolic gap between them and their work : it is then easier for them to view it differently.

The focus that advising puts on the activity of language learning and not on linguistic acquisition leads learners to give importance to aspects which they may tend to hand over to teachers or course book authors, thus becoming more aware of the control which they can have on their own learning process. But the fundamental element is that while talking to the adviser, learners are in fact talking to themselves. As we have just seen with Lorraine, learners are led to gradually internalize into their own internal cognitive processes, the reflexive elements which are structured in advising through an intersubjective process between learner and adviser. One can truly recognize here that learners become autonomous, that is, fully able to decide by themselves.

The advising session provides the conditions for what can be described as the conceptual or epistemological rupture that learners need in order to engage in the transformation of their representations and behaviours. In addition, as advising sessions are concerned with the reality of each learner"s specific learning experience, they allow learners to talk openly about their personal learning practices, some of which they may have kept hidden from their language teachers for fear of disapproval. Thus the verbal interaction with the adviser allows them to give legitimate existence to these practices.

One can then see how important it is that advisers and learners share a common language.

Using the learner"s mother tongue for advising9 is a deliberate choice in SDLLS. Talking about one"s learning activity is something highly technical for learners: they must feel sufficiently at ease to be able to express themselves fluently and appropriately. But efficiency is not the only justification for the choice of the mother tongue: it also relates to the fact that the relationship is not founded on power, as learners will not feel at a disadvantage in the interaction. It also makes the objective of the advising sessions quite explicit: through the use of their mother tongue, learners are thus made aware that the advising sessions are not work

Procedures». In F. Kjisik, P. Voller, N. Aoki and Y. Nakata (eds) Mapping the Terrain of Leaner Autonomy, Tampere, Tampere University Press, pp. 145-167.

13 sessions, and advisers clearly indicate that they do not use the advising sessions to evaluate the learner"s communicative competence in the foreign language.

One can deduce from this that the relational modalities for advising are complex10 . They require specific communicative competence and specific expertise in advisers, and specific conditions in the advising context: I will now discuss these three dimensions.

The communicative characteristics of advising

The advising conversation is a one-to-one communicative situation between a learner and an adviser11 which rests on the assumption that through a discussion about their learning procedures, learners will be able a) to transform the conceptions which underlie the decisions they make about learning a language, and b) to increase the methodological repertoire at their disposal when they actually work on learning the language. This is a very different situation from the teaching situation: the teacher-learner relationship is of a different nature as it links a group of learners to an individual teacher.

As I have already stated above, the difference between the adviser"s and the teacher"s role is reflected in the difference in the interactive modalities. In the classroom, the teacher"s role is to organize the learning activities: for example, the teacher verbally gives instructions, designates pupils, and motivates their involvement in the class activities. The teacher"s role is also about assessing pupil"s knowledge: he/she will verbally request answers to questions, and then confirm or reject these answers. Teachers also have to organize and regulate the group"s discourse: they will open and close exchanges, distribute turns and regulate interactive discipline (Coulthard, 1977).

In advising, as I have illustrated in the case study, the adviser"s role relies on different communicative practices. To trigger the explanation process which will lead learners to reflect on aspects of their learning competence they may want to change, advisers in particular have to reformulate the learner"s words into their own, "more expert" words, to supplement it with didactical information, give their opinion on what the learner says (that the learner may freely discuss), to make suggestions (that the learner may freely accept or reject) (Gremmo, 1995).

The essential element for advisers is not to take decisions concerning learning on the part of the learner, and to adopt a helping attitude. This determines very precisely the linguistic forms that the adviser will use. For example, to make sure that the suggestions they make will be understood as suggestions and not instructions, advisers tend to use attenuating intonation and highly-modalized forms such as: "perhaps you could..", "what could perhaps be interesting for you is..".

The organization of the interaction must also seek to reflect the greater equality that selfdirected learning implies between the learner"s and the adviser"s interactional statuses. This relates in particular to the control of the interaction. In the advising interactive modality, there is a contradiction between the didactic situation and the social conditions of the two participants. The didactic situation implies that it is the learners" role to monitor the interaction, as they are in charge of their learning and the adviser is there to help. This means that learners should for example open and close the session, or introduce conversational topics. However, the social conditions tip the scales towards the advisers. The advising sessions take place in their institution, they have professional expertise: common social rules imply that advisers have the right, even the duty, to control the interaction. So in my experience, the fact that learners take an active part in controlling the advising interaction (by actually opening and closing it, by introducing the discussion topics, as Lorraine did in the case study) is representative of their assuming the responsibility for their learning activity as a whole, and more specifically of their use of the advising session as a truly personal "helping scheme".

Thus, one function of the adviser"s interactive role is to bring learners to view themselves as the person in charge of the interaction. Advisers do so by openly explaining their respective roles. They also do so by gradually refusing to assume the management of the interaction, leaving their turn as soon as learners attempt to break in, or letting learners deal with the task of filling in the silences which, without exception, do occur. As one can see, the dialogical communicative competence that is in coherence with the characteristics of advising is quite specific, and the development of such a competence is one of the significant elements at stake in the training of advisers. Without this communicative change, advising conversations run the risk of remaining teaching in disguise in which, because of the "natural" asymmetry of the learner/adviser relationship, advisers would, more or less unconsciously, impose their decisions, even with the best of intentions. In a self-directed learning context, the coherence between the adviser"s role and the adviser"s language necessarily implies that the advising conversation should truly be a negotiative situation which enables learners to give it meaning, and in which advisers are fully aware that they only have part of the information and that they cannot pre-determine what learners will achieve.

The nature of the adviser's expertise

Throughout this article, I have used the term "advising conversation" to refer to the interactive modality. It is now time to point out that the term "conversation" here is not to be understood as "small talk" or "friendly" conversation. The term conversation here is used to differentiate advising from teaching, and to emphasize the fact that advising is a one-to-one relationship where the two participants have equal status. But it is also very important to understand that an advising conversation has a specific structure which makes it a "professional" situation.

The advising conversation has a well-defined focus and refers to a well-defined domain of both knowledge and know-how. In addition to the specific communicative competence I have just discussed, the expertise of advisers also deals with other areas of professional competence. Advisers are competent at analyzing the learner"s contribution to the discourse using the specific frame of reference of foreign language didactics, as this scientific field both defines and delimits their field of action. They are competent at analyzing what the learners say in order to indicate inaccuracies or gaps in the learner"s set of mental representations, cognitive sets and methodological procedures as far as they refer to foreign language and foreign language learning. They are also competent to provide information, suggest procedures, and describe learning activities which learners would not have thought of alone.

Their contribution essentially, if not exclusively, consists of helping learners to develop their language learning competence and to become more autonomous learners. It follows that the adviser"s expertise comprises:

 detailed science-based knowledge about the nature of language, about the concepts which organize this field of reference and their evolution, as well as the methodological know-how referring to the didactic methodologies which constitute common knowledge; 12  detailed science-based knowledge of the nature of the language learning process, and especially of the nature of self-directed learning and its implications for the learner;

 detailed knowledge of the specific SDLLS in which the adviser is working, including detailed knowledge of the resources available. 13 One can thus see that the adviser"s expertise is multidisciplinary, since it comprises theoretical methodological aspects of various fields of linguistics 14 , psychology, and language didactics. This expertise determines the quality of the mental analysis that advisers will make during an advising conversation and that of the contributions which they will offer. This specific expertise is thus a fundamental factor for the success of SDLLS.

Specific conditions in the advising context

The conditions in which advising sessions take place can have an important influence on learners" understanding of their role. First of all, it is crucial that advising sessions should be given institutional existence. Thought must be given to the material conditions in which the sessions will function: a specific place furnished with the conversational objective in view, pleasant surroundings, and privacy, are signals given to learners that the moment is seen as relevant by the institution.

Advising can only play its role fully if it is inserted between two work sessions on the learner"s part. It seems to me very important that learners should be aware from the very beginning that there will be an organized alternation of periods of work and regular "reflexive 12 This means that advisers are not as much specialized in one specific methodology as they are able to present learners with various approaches, discussing their methodological proposals, techniques and types of activities. 13 For example: types of resources, types of equipment, etc. Again the important point is to be able to discuss the rationale of the various options which the SDLLS makes available. 14 For example, sociolinguistics, psycholinguistics, general linguistics, descriptive linguistics of the various languages concerned, etc.

intervals" of a different nature from their working sessions, so that that they consider the advising sessions as an integral part of the SDLLS .

The advising sessions also benefit when they can take place within a sufficient time space: the advising relationship has developmental objectives which can only be carried out with a certain time framework, so that advising can be more than just solving material problems. On the other hand, it is also necessary to mark the limits within which it is organized and to clearly explain to learners what time space is available for them.

My last remark deals with the importance of a relational continuity. The advising relationship is a changing relationship and, in my case study, I have shown how the adviser has to balance her action between imposition and laissez-faire in differentiated ways according to her understanding of the learner"s evolution. On the other hand, learners need time to build up the communicative routines that will make it gradually easier for them first to take part in, and then to take control of the advising conversation. There is thus a need for the setting-up of a "permanent" interactive learner/adviser pair that will ensure that a discursive, learning and relational history is established to allow for the development of the learner"s autonomy.

  She saw the advising session in terms of an expert/non-expert relationship and she had trust in the adviser"s expertise:

	17. work took place when I was alone at home (…) the advising sessions gave me help,
	gave me ideas for my work, small bits of advice (…) I never felt I had to follow them
	(…)

18. the advice she gives is generally useful because she knows what she is talking about. I would test the things she suggested to see how I could use them

But her own conclusion can be judged at first as rather disappointing:

19. the advising sessions helped me but they did not really change the way I work

Nancy, for her part, seemed to share Lorraine"s point of view. She felt that 20. Lorraine very much kept to her own criteria, she changed a little but not much,

  Procedures». In F. Kjisik, P. Voller, N. Aoki and Y. Nakata (eds) Mapping the Terrain of Leaner Autonomy, Tampere, Tampere University Press, pp. 145-167.

Procedures». In F. Kjisik, P. Voller, N. Aoki and Y. Nakata (eds) Mapping the Terrain of Leaner

Procedures». In F. Kjisik, P. Voller, N. Aoki and Y. Nakata (eds) Mapping the Terrain of Leaner Autonomy, Tampere, Tampere University Press, pp. 145-167.

[START_REF] Ciekanski | L'accompagnement à l'autoformation en langue étrangère: contribution à l'analyse des pratiques professionnelles[END_REF]. The SDLLS concerned is the SDLLS set up at C.R.A.P.E.L. (Nancy, France) for French adults learning English as a foreign language.

I will return to this important point in the last section of the article.

All quotations from the advising sessions have been translated into English. They are originally in French.

As presented above: firstly, the recordings of the advising sessions and secondly, the recordings of the interviews with Lorraine and Nancy. .

A "low quality" learning activity could for example refer to the situation in which a learner feels he/she did not work sufficiently. He/she then may feel that there is "nothing to talk about" but talking about why he/she could not work sufficiently is of value for the development of his/her self-directed learning competence.

This entails that the adviser feels fluent in the learner"s mother tongue. When the choice of the learner"s mother tongue is not possible, the advising conversation may be in a common language in which both learner and adviser feel fluent. Otherwise, there can be no advising conversation as such.

Nancy"s case shows that even an experienced adviser can get the impression that she is not successful, while at the same time objective data can prove that she does have a very positive influence.

A few SDLLS sometimes offer advising sessions for small groups of learners. ,


Cédric Buche
email: [buche@enib.fr

Ronan Querrec
email: querrec@enib.fr

Eric Maffre
email: maffre@enib.fr

Pierre Chevaillier
email: chevaillier@enib.fr

Pierre Deloor
email: deloor]@enib.fr

MASCARET: Multi-Agents System for Virtual Environment for Training

Keywords: Virtual Environment for Training, Multi-Agents Systems, Collaborative Work, Pedagogy. 1 Introduction

This study concerns virtual environments for training in operational conditions. The principal developed idea is that these environments are heterogeneous and open multi-agent systems. The MASCARET model is proposed to organize the interactions between agents and to provide them reactive, cognitive and social abilities to simulate the physical and social environment. The physical environment represents, in a realistic way, the phenomena that learners and teachers have to take into account. The social environment is simulated by agents executing collaborative and adaptive tasks. These agents realize, in team, procedures that they have to adapt to the environment. The users participate to the training environment through their avatar. In this article, we show that MASCARET allows the establishment of models necessary to the creation of Intelligent Tutoring System. We interest notably to its use in pedagogical aspects.

This study concerns virtual environments for training in operationnal conditions. We want to simulate and to immerge the learners in their professional environment. This enables them to manipulate the environment so that they can "learn while doing". This idea is driven by the "constructivism" paradigm defined by Piaget [START_REF] Piaget | Le comportement, moteur de l'évolution[END_REF] and can find a good implementation in virtual reality technics as presented by [START_REF] Fuchs | Le traité de la réalité virtuelle[END_REF][START_REF] Burdea | La réalité virtuelle[END_REF]. Our definition of Virtual Reality is the one proposed by [START_REF] Tisseau | Réalité Virtuelle : autonomie in virtuo[END_REF] which proposes to give autonomy to models involving in the virtual environment by giving them the "triple mediation of sens, decision and action". So, the main developed idea is that virtual environments for training are heterogeneous and open multi-agent systems. Those Multi-Agents Systems (MAS) has been presented by [START_REF] Demazeau | From interactions to collective behaviour in agent-based systems[END_REF] using the VOWELS model considering a MAS with four vowels: Agent, Environment, Interaction and Organisation. From our point of view, we consider the user of a virtual environment as other autonomous agents because he can interact with the environment and with other agents or users in the same way. Then, as [START_REF] Tisseau | Réalité Virtuelle : autonomie in virtuo[END_REF], we propose to add a last vowel, the letter U for User, in the VOWELS model.

Our perspective is to create an Intelligent Tutoring System (ITS). Its aims at providing students with dedicated tutoring. Its goal is to communicate knowledge effectively. ITS can be composed of fourth components. First, the model of the domain is the representation of the teacher's expertise. Second, the model of the learner allows knowing at each instant the state of the student's knowledge. Then, the pedagogical model refers to the methods of instruction. Using student's behaviour and his model, the pedagogical model allows selecting learning strategies. Finally, the interface model permits the communication between the system and the student.

Our objective is to place learners in operational positions in their simulated physical and social environment. We show the MASCARET model, proposed to organize the interactions between agents and to provide them reactive, cognitive and social abilities to simulate the physical and social environment. Then we see that MASCARET offers mechanisms to implement the functions present in existing ITS. We interest notably to its use in pedagogical aspects. Finally, we conclude and envisage futures work.

The MASCARET model.

Our goal is to train teams to collaborative and procedural work in a physical environment. In this case, we have to simulate in a realistic way this physical environment and the collaborative and adaptive team member's behavior in the social environment. Evolution of those environments results from simulation of autonomous agent's local behavior and their interactions. We propose a model call MASCARET where we use multi-agents systems to simulate realistics, collaboratives and adaptives environments for training. This model aims at organizing the interactions between agents and provides them abilities to evolve in this context.

2.1

The organisational model.

As the users has to be integrated both in the social environment (member of a particular team) and in the physical environment (to undergo a lick of gaz for exemple), we propose, first a generic organisational model allowing to represent the physical and the social environment. The model we propose is founded upon the concepts of organisation, roles, behavioral features and agents (Figure 1). [START_REF] Hannoun | MOISE : Un modèle organisationnel pour la conception de systèmes multi-agents[END_REF] has already proposed an organisational model for multiagents systems, but this model, dedicated to the collaborative realisation of procedures, is not enougth generic to solve our problem. [START_REF] Gutknecht | Vers une méthodologie organisationnelle de conception de systèmes multi-agents[END_REF] has also proposed such a model called Agent/Group/Role, but this model seems to be more a pattern for conception than a model which really formalizes the concepts of organisation and roles. In our model, the aim of the organisation is to structure interactions between agents; it enables each agent to know its partners and the role they are playing in the collaboration. The concept of role represents the responsabilities (behavioral features) played by agents in the organisation. Agents have then an organisational behavior which permits them to play or abandon a role in a organisation. This behavior enables also agents to take into account the existence of other members. This model is a generic model in the way that all the resulting classes are abstract. The organisational model is then derived to implement two concrets organisations representing physical and social environment that have to be simulated in the virtual environment for training.

Figure 1: The organisational model.

2.2

The physical environment.

In a virtual environment for training, the users's (learner and teacher) physical environment must be realistic, interactive and act in "real-time". Then, to reach the constraints of virtual reality, models we use to simulate physical phenomena are obiously simplified. Therefore, the teacher may wants, for pedagogical reasons, inhibates some phenomena. For that, we must propose models which are compatible to a disconnection between the phenomena. Moreover, althought all interactions have potentially effects on the two agent involved, in most cases the effect of one agent is more important. We consider then that the interactions between agents have a privilieged direction. Then, the reactive agents' behaviour evolving in physical environment is to percieve situations where there is interactions and to act consequensly. A practical limit of the individual based models is that each agent can potentially percieve all others.

The complexity of the algorithm is in this case O(n 2 ). Then, we have to design rules to organize thoses interactions between reactive agents. For that, we use then the generic organisational model we have proposed before. The organisation is, in this case, a network where agents are connected together when they are in interaction. We call this organisation an interaction network (InteractionNet, Figure 2). To represent the concept of privilieged direction in interactions, we define two particular roles called source and target. The goal of source agents is to give information on their internal states to other agents (targets) so that they can compute the interaction's value and their internal state. The interaction can be detected by the two agents involved, but, for "real-time" computation reasons, it's better if only one agent detect it (one of two agents or another one else). We then define a recruiting role which has the responsability to maintain the knowlege of each agent upon the structure of the organisation. This means that an agent playing this role has to detect the interactions situations. The internal architecture of reactive agents match the constraint of physical phenomena disconnection presented before, because an agent can have several reactive behavior, each one participating in a different interaction network. This elementary behavior consist in the computation of a vector of internal state variables after the evaluation of inputs (from the interactions where the agent is a target) and presents a pertinent internal state to other agents (potentialy targets of an interaction where the former agent is a source).

Figure 2: The Interactions Network.

2.3

The social environment.

The social environment is also populated by more "intelligent" agents. They undergo it and they act on it as reactive agents, but the way they choose their actions is carried out on a higther level of abstraction. Those agents are various humans involving in the formation (learners and teachers) who are played by autonomous agents. In our case, the social environment is structured and each member knows its roles and those of its partners. The interactions between the team members are also structured and arranged by the mean of procedure known by all members. We thus derive our generic organisational model to formalize this concept of team. We are interested in the case where the action's coordination between team members are already envisaged and written in procedure. On the other hand, the environment being dynamic, agents can need to adapt the scenario to the environment. The procedure must then have a semantic representation so that agents can reason above. To describe a procedure we use the temporal logic of Allen (logic on the intervals of time).

The reasoning of team members relates on organisation, procedures and actions. We propose a model of agent having local organisational knwoledge. An agent is divided into a decisional part and a part represented by modules of perception of the physical environment, communication and actions (Figure 3). The agent must carry out actions of the procedure and adapt to situations not envisaged. The procedure describes interactions between agents in an optimal case, and leaves to the agent the responsability to build implicit plans (not clarified in the procedure) considered as natural within an applicative situation. Moreover, the procedure organize actions of a semantic level which we call « actions trades » such as « sprinkling a fire » in the case of firemen procedures, whereas the implicit plans arrange actions of a generic semantic level for humans such as « going at a point ». For that, the agent must be able to reason on actions and we propose a model of goal directed actions having pre-conditions and post-conditions. Thus, before carrying out an action, the agent must make sure that pre-conditions of this one are checked. If it is not the case, it builds itself a plan by back chaining on pre-conditions and post-conditions of actions. When an agent starts or stops actions, it broadcasts a message that enables other members to follow the evolution of the procedure. When this behavior is at fault, the agent calls it's organisational behavior which can help it to find a solution with another team member. Thus, in a hierarchical organisation, when an agent has a problem which it cannot solve, it refers to its superior. Then, the superior has the responsability to find a solution among his subordinates. If it does not find any, it refers to its own superior about it. We represent this mechanism by a method like a Contract Net Protocol.

2.4

The users.

The avatar is the representation of the user in the environment. The model of avatar we propose is the same as rational agents except for the inhibition of some messages. Indeed, all modules composing the avatar are active and thus remain potentialy usable; however certain communication are desactiveted. Thus the collaborative behavior does not call any more the organisational behavior (in case of a failure in an action) and the reasoning module does not communicate with the operative part to ask it to execute actions. These decisions are from the responsability of the learner. This model enables the avatar to follow the evolution of the procedure and the choice of the users. Having this capacity, the avatar of a learner can explain advice or show the realization of a task to the user.

MASCARET and Intelligent

Tutoring Systems.

Intelligent Tutoring System (ITS) aims at providing students with dedicated tutoring. We use ITS as a system allowing to communicate a knowhow. ITS can be composed of fourth components: model of the domain, model of the student, pedagogical model and model of the interface. It usually uses only two models of those fourth components (domain and student). Pedagogical and interface model is not often well specified.

MASCARET allows the establishment of models necessary to the creation of Intelligent Tutoring System. Concerning the model of the domain, we propose as [START_REF] Lourdeaux | Réalité virtuelle et formation : conception d'environnement virtuels pédagogiques[END_REF] the use of the notion of procedure as describe in the social environment. Thanks to this formalism, an expert is able to provide such a model using semantics and temporal logic. There is no explicit modelling of the student in MASCARET. But the model of the student could be obtained using active detection of the student's actions and comparing their to the model of the domain.

The pedagogical model could take decision using the model of the domain and the model of the student. First, we propose to use the physical environment of MASCARET in a pedagogical way. In fact, thanks to different level of interaction network, physical phenomena could be adapted to the student level. It is not necessary to display every disturbing elements for a novice. As opposite, it is possible to add new physical disturbing phenomena for an experimented student in order to improve his skill. The physical environment of MASCARET also allows showing elements that are not visible in the real [START_REF] Mellet D'huart | Virtual reality for education and training: State of science and typology of uses[END_REF]. For example, we could display wind curve or gasses cloud in order to specify to the student specific conditions. In addition, the architecture of rational agents can supply to the user the action plan needed to realize a task. Using the notion of procedure as describe in the social environment, the avatar can provide deductive reasoning and explications [START_REF] Rickel | Animated agents for procedural training in virtual reality: Perception, cognition, and motor control[END_REF]. In fact, the ITS reply to answers how and why something must be done during the exercise. The ITS can also reply to what should be done next and therefore a simulated tutor can show a demonstration of the next action in line. Finally, we propose the use of MASCARET to define the organisation of pedagogical role in a multi strategic pedagogical teaching.

Our perspective is to integrate pedagogical actors, taking different roles. Therefore such actors have a common objective: to increase the student's skills.

Conclusion and future works.

Our objective is to place learners in operational positions in their simulated physical and social environment.

Considering a virtual environment for training as a multi-agents system, we propose the model MASCARET. It allows the realization of a realistic, cooperative and adaptative virtual environment for training. In MASCARET, classical pedagogical functionalities expected in a virtual environment for training have not been formalized.

The study of different training environments showed that MASCARET allows the implementation of the functions present in existing environments and offers mechanisms allowing them to be improved. Thus, we wish to integrate the notion of "pedagogical program" proposed by pedagogues [START_REF] Richard | Les activités mentales. Comprendre, raisonner, trouver des solutions[END_REF]. It will be built around three phases: 1) the verbalisation, the learner is able to explain the objectives of the formation, 2) the transfer, the learner is able to abstract concepts present in the training and to transfer them to another context and 3) the reinforcement, the learner effectuates a series of exercices aiming to create automatism on the subject of the training. Our contribution will concern the realization of agent models allowing us to automatically distinguish these three phases in the learner program and to generate adequate exercises.

In order to improve the learner apppropriation of his role in the exercice, we also propose the integration of the notion of "putting into operation" and of "pedagogical scenario" [START_REF] Crampes | Facteurs qualité et composantes de scénario pour la conception de simulateurs pédagogiques à vocation comportementale[END_REF]. That way the learner will feel immersed in the context of the exercice and decides by himself to play a role. Thus exercise of the "program" begins by a phase of putting into operation of a pedagogical context. In addition, virtual reality and multi-agents systems allow the simulation of different elements (characters, physical phenomena…) allowing to trigger learner emotions, that is an important factor in pedagogy. Thus every exercise of the "program" begins by a phase of putting into operation of a pedagogical context.

We envisage the use, during the exercise, different strategies of pedagogical learning. For that, we propose to model pedagogical strategies by means of pedagogical actors. In this framework, we noticed the strategy of the critic, counselor, guardian, companion [START_REF] Chan | Learning Compagnion Systems[END_REF] and troublemaker [START_REF] Aimeur | Cooperative learning strategies for intelligent tutoring systems[END_REF]. We are particularly interested in the cooperative strategy, where the companion is a virtual actor that will cooperate for the realization of tasks, exchange ideas on the problem and share the same goals. We also are interested in the troublemaker strategy where the goal is to disturb the learner by proposing solutions that can sometimes be erroneous. That way, we force the learner to evaluate his self-confidence in his own solutions. We propose to the human teacher, means the human expert, to specify different pedagogical agent behaviors (companion, troublemaker ....). For that, we wish to endow such agents with the capacity of learning by imitation [START_REF] Gaussier | From perception-action loops to imitation processes: A bottom-up approach of learning by imitation[END_REF] or by the exemple [START_REF] Del Bimbo | Specification by-example of virtual agents behavior[END_REF]. The human teacher will take control of the pedagogical agents during the preparation phase of the exercice and they will learn a behavior adapted to the situation. If it is considered we are in possession of such different pedagogical agents, an ITS will be multi strategic [START_REF] Mengelle | A multi-Agent Architecture for an ITS with multiple strategies[END_REF]. That will necessitate a selection mechanism of strategy regarding the context (learner level, pedagogical opportunities ….).

Figure 3 :

 3 Figure 3: Architecture of rational agents.


Ivan Nourdin

Giovanni Peccati

Roland Speicher

Multidimensional semicircular limits on the free Wigner chaos

Keywords: Convergence in Distribution, Fourth Moment Condition, Free Brownian Motion, Free Probability, Multidimensional Limit Theorems, Semicircular Law, Wigner Chaos. 2000 Mathematics Subject Classification: 46L54, 60H05, 60H07, 60H30

.

Introduction

Let W = {W t : t ≥ 0} be a one-dimensional standard Brownian motion (living on some probability space (Ω, F , P )). For every n ≥ 1 and every realvalued, symmetric and square-integrable function f ∈ L 2 (R n + ), we denote by I W (f ) the multiple Wiener-Itô integral of f , with respect to W . Random variables of this type compose the so-called nth Wiener chaos associated with f . In an infinite-dimensional setting, the concept of Wiener chaos plays the same role as that of the Hermite polynomials for the one-dimensional Gaussian distribution, and represents one of the staples of modern Gaussian analysis (see e.g. [START_REF] Janson | Gaussian Hilbert Spaces[END_REF][START_REF] Nourdin | Normal Approximations using Malliavin Calculus: from Stein's Method to Universality[END_REF][START_REF] Nualart | The Malliavin calculus and related topics[END_REF][START_REF] Peccati | Wiener Chaos: Moments, Cumulants and Diagrams[END_REF] for an introduction to these topics).

In recent years, many efforts have been made in order to characterize Central Limit Theorems (CLTs) -that is, limit theorems involving convergence in distribution to a Gaussian element -for random variables living inside a Wiener chaos. The following statement gathers the main findings of [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF] (Part 1) and [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF] (Part 2), and provides a complete characterization of (both one-and multi-dimensional) CLTs on the Wiener chaos.

Theorem 1.1 (See [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF][START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF]).

(A) Let F k = I W (f k ), k ≥ 1, be a sequence of multiple integrals of order n ≥ 2, such that E[F 2 k ] → 1. Then, the following two assertions are equivalent, as k → ∞: (i) F k converges in distribution to a standard Gaussian random variable N ∼ N (0, 1);

(ii) E[F 4 k ] → 3 = E[N 4 ]. 1
Université Nancy 1, France. Email: inourdin@gmail.com 2 Université du Luxembourg, Luxembourg. Email: giovanni.peccati@gmail.com 3 Universität des Saarlandes, Germany. Email: speicher@math.uni-sb.de 1 (B) Let d ≥ 2 and n 1 , ..., n d be integers, and let (F

k , ..., F

k ), k ≥ 1, be a sequence of random vectors such that, for every i = 1, ..., d, the random variable F (i) k lives in the n i th Wiener chaos of W . Assume that, as k → ∞ and for every i, j = 1, ..., d, E[F

(i) k F (j) k ] → c(i, j)
, where c = {c(i, j) : i, j = 1, ..., d} is a positive definite symmetric matrix. Then, the following two assertions are equivalent, as k → ∞:

(i) (F (1) k , ..., F (d) k ) converges in distribution to a centered d-dimensional Gaussian vector (N 1 , ..., N d ) with covariance c; (ii) for every i = 1, ..., d, F (i) k
converges in distribution to a centered Gaussian random variable with variance c(i, i).

Roughly speaking, Part (B) of the previous statement means that, for vectors of random variables living inside some fixed Wiener chaoses, componentwise convergence to Gaussian always implies joint convergence. The combination of Part (A) and Part (B) of Theorem 1.1 represents a powerful simplification of the so-called 'method of moments and cumulants' (see e.g. [START_REF] Peccati | Wiener Chaos: Moments, Cumulants and Diagrams[END_REF]Chapter 11] for a discussion of this point), and has triggered a considerable number of applications, refinements and generalizations, ranging from Stein's method to analysis on homogenous spaces, random matrices and fractional processes -see the survey [START_REF] Nourdin | Stein's method meets Malliavin calculus: a short survey with new estimates[END_REF] as well as the forthcoming monograph [START_REF] Nourdin | Normal Approximations using Malliavin Calculus: from Stein's Method to Universality[END_REF] for details and references. Now, let (A , ϕ) be a non-commutative tracial W * -probability space (in particular, A is a von Neumann algebra and ϕ is a trace -se Section 2.1 for details), and let S = {S t : t ≥ 0} be a free Brownian motion defined on it. It is well-known (see e.g. [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and ananlysis on Wigner space[END_REF]) that, for every n ≥ 1 and every f ∈ L 2 (R n + ), one can define a free multiple stochastic integral with respect to f . Such an object is usually denoted by I S (f ). Multiple integrals of order n with respect to S compose the so-called nth Wigner chaos associated with S. Wigner chaoses play a fundamental role in free stochastic analysis -see again [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and ananlysis on Wigner space[END_REF].

The following theorem, which is the main result of [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF], is the exact free analogous of Part (A) of Theorem 1.1. Note that the value 2 coincides with the fourth moment of the standard semicircular distribution S(0, 1). Theorem 1.2 (See [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF]). Let n ≥ 2 be an integer, and let (f k ) k∈N be a sequence of mirror symmetric (see Section 2.2 for definitions) functions in

L 2 (R n + ), each with f k L 2 (R n + ) = 1.
The following statements are equivalent. [START_REF] Biane | Free hypercontractivity[END_REF] The fourth moments of the stochastic integrals

I(f k ) converge to 2, that is, lim k→∞ ϕ(I S (f k ) 4 ) = 2.
(2) The random variables I S (f k ) converge in law to the standard semicircular distribution S(0, 1) as k → ∞.

The aim of this paper is to provide a complete proof of the following Theorem 1.3, which represents a free analogous of Part (B) of Theorem 1.1.

Theorem 1.3. Let d ≥ 2 and n 1 , . . . , n d be some fixed integers, and consider a positive definite symmetric matrix c = {c(i, j) : i, j = 1, ..., d}. Let (s 1 , . . . , s d ) be a semicircular family with covariance c (see Definition 2.10). For each i = 1, . . . , d, we consider a sequence (f

(i) k ) k∈N of mirror-symmetric functions in L 2 (R n i +
) such that, for all i, j = 1, . . . , d,

(1.1) lim k→∞ ϕ[I S (f (i) k )I S (f (j) k )] = c(i, j).
The following three statements are equivalent as k → ∞.

(1) The vector ((I S (f

k ), . . . , I S (f

(d) k )) converges in distribution to (s 1 , . . . , s d ).
(2) For each i = 1, . . . , d, the random variable I S (f

(i) k ) converges in distri- bution to s i . (3) For each i = 1, . . . , d, lim k→∞ ϕ[I S (f (i) k ) 4 ] = 2 c(i, i) 2 .
Remark 1.4. In the previous statement, the quantity ϕ[I S (f

(i) k )I S (f (j) k )] equals f (i) k , f (j) k L 2 (R n i + ) if n i = n j ,
and equals 0 if n i = n j . In particular, the limit covariance matrix c is necessarily such that c(i, j) = 0 whenever n i = n j . Remark 1.5. Two additional references deal with non-semicircular limit theorems inside the free Wigner chaos. In [START_REF] Nourdin | Poisson approximations on the free Wigner chaos[END_REF], one can find necessary and sufficient conditions for the convergence towards the so-called Marčenko-Pastur distribution (mirroring analogous findings in the classical setting -see [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF]). In [START_REF] Deya | Convergence of Wigner integrals to the tetilla law[END_REF], conditions are established for the convergence towards the so-called 'tetilla law' (or 'symmetric Poisson distribution' -see also [START_REF] Nica | Commutators of free random variables[END_REF]).

Combining the content of Theorem 1.3 with those in [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF][START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF], we can finally state the following Wiener-Wigner transfer principle, establishing an equivalence between multidimensional limit theorems on the classical and free chaoses.

Theorem 1.6. Let d ≥ 1 and n 1 , . . . , n d be some fixed integers, and consider a positive definite symmetric matrix c = {c(i, j) : i, j = 1, ..., d}. Let (N 1 , . . . , N d ) be a d-dimensional Gaussian vector and (s 1 , . . . , s d ) be a semicircular family, both with covariance c. For each i = 1, . . . , d, we consider a sequence (f

(i) k ) k∈N of fully-symmetric functions (cf. Definition 2.2) in L 2 (R n i + ). Then: (1) For all i, j = 1, . . . , d and as k → ∞, ϕ[I S (f (i) k )I S (f (j) k )] → c(i, j) if and only if E[I W (f (i) k )I W (f (j) k )] → (n i )!(n j )! c(i, j).
(2) If the asymptotic relations in (1) are verified then, as k → ∞,

I S (f (1) k ), . . . , I S (f (d) k ) law → (s 1 , . . . , s d )
if and only if

I W (f (1) k ), . . . , I W (f (d) k ) law → (n i )!N 1 , . . . , (n d )!N d .
The remainder of this paper is organized as follows. Section 2 gives concise background and notation for the free probability setting. Theorems 1.3 and 1.6 are then proved in Section 3.

Relevant definitions and notations

We recall some relevant notions and definitions from free stochastic analysis. For more details, we refer the reader to [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and ananlysis on Wigner space[END_REF][START_REF] Kemp | Wigner chaos and the fourth moment[END_REF][START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF].

2.1. Free probability, free Brownian motion and stochastic integrals. In this note, we consider as given a so-called (tracial) W * probability space (A , ϕ), where A is a von Neumann algebra (with involution X → X * ), and ϕ : → C is a tracial state (or trace). In particular, ϕ is weakly continuous, positive (that is, ϕ(Y ) ≥ 0 whenever Y is a nonnegative element of A ), faithful (that is, ϕ(Y Y * ) = 0 implies Y = 0, for every Y ∈ A ) and tracial (that is, ϕ(XY ) = ϕ(Y X), for every X, Y ∈ A ). The self-adjoint elements of A are referred to as random variables. The law of a random variable X is the unique Borel measure on R having the same moments as X (see [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Proposition 3.13]). For 1 ≤ p ≤ ∞, one writes L p (A , ϕ) to indicate the L p space obtained as the completion of A with respect to the norm a p = τ (|a| p ) 1/p , where |a| = √ a * a, and • ∞ stands for the operator norm.

Definition 2.1. Let A 1 , . . . , A n be unital subalgebras of A . Let X 1 , . . . , X m be elements chosen from among the A i 's such that, for 1 ≤ j < m, X j and X j+1 do not come from the same A i , and such that ϕ(X j ) = 0 for each j. The subalgebras A 1 , . . . , A n are said to be free or freely independent if, in this circumstance, ϕ(X 1 X 2 • • • X n ) = 0. Random variables are called freely independent if the unital algebras they generate are freely independent. Definition 2.2. The (centered) semicircular distribution (or Wigner law) S(0, t) is the probability distribution

(2.1) S(0, t)(dx) = 1 2πt 4t -x 2 dx, |x| ≤ 2 √ t.
Being symmetric around 0, the odd moments of this distribution are all 0. Simple calculations (see e.g. [7, Lecture 2]) show that the even moments can be expressed in therms of the so-called Catalan numbers: for non-negative integers m,

2 √ t -2 √ t x 2m S(0, t)(dx) = C m t m ,
where

C m = 1 m+1 2m
m is the mth Catalan number. In particular, the second moment (and variance) is t while the fourth moment is 2t 2 . Definition 2.3. A free Brownian motion S consists of: (i) a filtration {A t : t ≥ 0} of von Neumann sub-algebras of A (in particular, A s ⊂ A t , for 0 ≤ s < t), (ii) a collection S = {S t : t ≥ 0} of self-adjoint operators in A such that: (a) S 0 = 0 and S t ∈ A t for every t, (b) for every t, S t has a semicircular distribution with mean zero and variance t, and (c) for every 0 ≤ u < t, the increment S t -S u is free with respect to A u , and has a semicircular distribution with mean zero and variance tu.

For the rest of the paper, we consider that the W * -probability space (A , ϕ) is endowed with a free Brownian motion S. For every integer n ≥ 1, the collection of all operators having the form of a multiple integral

I S (f ), f ∈ L 2 (R n + ; C) = L 2 (R n + ), is defined according to [2, Section 5.3], namely: (a) first define I S (f ) = (S b 1 -S a 1 ) • • • (S bn -S an ) for every function f having the form (2.2) f (t 1 , ..., t n ) = 1 (a 1 ,b 1 ) (t 1 ) × . . . × 1 (an,bn) (t n ),
where the intervals (a i , b i ), i = 1, ..., n, are pairwise disjoint; (b) extend linearly the definition of I S (f ) to 'simple functions vanishing on diagonals', that is, to functions f that are finite linear combinations of indicators of the type (2.2); (c) exploit the isometric relation

(2.3) I S (f ), I S (g) L 2 (A ,ϕ) = R n + f (t 1 , . . . , t n )g(t n , . . . , t 1 )dt 1 . . . dt n ,
where f, g are simple functions vanishing on diagonals, and use a density argument to define I(f ) for a general f ∈ L 2 (R n + ). As recalled in the Introduction, for n ≥ 1, the collection of all random variables of the type I S (f ), f ∈ L 2 (R n + ), is called the nth Wigner chaos associated with S. One customarily writes I S (a) = a for every complex number a, that is, the Wigner chaos of order 0 coincides with C. Observe that (2.3) together with the above sketched construction imply that, for every n, m ≥ 0, and every

f ∈ L 2 (R n + ), g ∈ L 2 (R m + ), (2.4) ϕ[I S (f )I S (g)] = 1 n=m × R n + f (t 1 , . . . , t n )g(t n , . . . , t 1 )dt 1 . . . dt n ,
where the right hand side of the previous expression coincides by convention with the inner product in L 2 (R 0 + ) = C whenever m = n = 0.

Mirror Symmetric Functions and Contractions.

Definition 2.4. Let n be a natural number, and let f be a function in

L 2 (R n + ). (1) The adjoint of f is the function f * (t 1 , . . . , t n ) = f (t n , . . . , t 1 ). (2) f is called mirror symmetric if f = f * , i.e. if f (t 1 , . . . , t n ) = f (t n , . . . , t 1 )
for almost all t 1 , . . . , t n ≥ 0 with respect to the product Lebesgue measure (3) f is called fully symmetric if it is real-valued and, for any permutation σ in the symmetric group Σ n , f (t 1 , . . . , t n ) = f (t σ(1) , . . . , t σ(n) ) for almost every t 1 , . . . , t n ≥ 0 with respect to the product Lebesgue measure.

An operator of the type I S (f ) is self-adjoint if and only if f is mirror symmetric.

Definition 2.5. Let n, m be natural numbers, and let f ∈ L 2 (R n + ) and g ∈ L 2 (R m + ). Let p ≤ min{n, m} be a natural number. The pth contraction f p ⌢ g of f and g is the L 2 (R n+m-2p + ) function defined by nested integration of the middle p variables in f ⊗ g:

f p ⌢ g (t 1 , . . . , t n+m-2p ) = R p + f (t 1 , . . . , t n-p , s 1 , . . . , s p )g(s p , . . . , s 1 , t n-p+1 , . . . , t n+m-2p ) ds 1 • • • ds p .
Notice that when p = 0, there is no integration, just the products of f and g with disjoint arguments; in other words, f 0 ⌢ g = f ⊗ g. i.e. min B i < min B j iff i < j. If each block consists of two elements, then we call the partition a pairing. The set of all partitions on [n] is denoted P(n), and the subset of all pairings is P 2 (n). Definition 2.6. Let π ∈ P(n) be a partition of [n]. We say π has a crossing if there are two distinct blocks B 1 , B 2 in π with elements x 1 , y 1 ∈ B 1 and

Non-crossing

x 2 , y 2 ∈ B 2 such that x 1 < x 2 < y 1 < y 2 .
If π ∈ P(n) has no crossings, it is said to be a non-crossing partition.

The set of non-crossing partitions of

[n] is denoted N C(n). The subset of non-crossing pairings is denoted N C 2 (n). Definition 2.7. Let n 1 , . . . , n r be positive integers with n = n 1 + • • • + n r . The set [n] is then partitioned accordingly as [n] = B 1 ⊔ • • • ⊔ B r where B 1 = {1, . . . , n 1 }, B 2 = {n 1 + 1, . . . , n 1 + n 2 }, and so forth through B r = {n 1 + • • • + n r-1 + 1, . . . , n 1 + • • • + n r }. Denote this partition as n 1 ⊗ • • • ⊗ n r .
We say that a pairing π ∈ P 2 (n) respects n 1 ⊗ • • • ⊗ n r if no block of π contains more than one element from any given block of

n 1 ⊗ • • • ⊗ n r . The set of such respectful pairings is denoted P 2 (n 1 ⊗ • • • ⊗ n r ). The set of non- crossing pairings that respect n 1 ⊗ • • • ⊗ n r is denoted N C 2 (n 1 ⊗ • • • ⊗ n r ).
Definition 2.8. Let n 1 , . . . , n r be positive integers, and let π ∈ P

2 (n 1 ⊗ • • • ⊗ n r ). Let B 1 , B 2 be two blocks in n 1 ⊗ • • • ⊗ n r . Say that π links B 1 and B 2 if there is a block {i, j} ∈ π such that i ∈ B 1 and j ∈ B 2 .
Define a graph C π whose vertices are the blocks of n 1 ⊗ • • • ⊗ n r ; C π has an edge between B 1 and B 2 iff π links B 1 and B 2 . Say that π is connected with respect to

n 1 ⊗ • • • ⊗ n r (or that π connects the blocks of n 1 ⊗ • • • ⊗ n r ) if the graph C π is connected. We shall denote by N C c 2 (n 1 ⊗ • • • ⊗ n r )
the set of all non-crossing pairings that both respect and connect

n 1 ⊗ • • • ⊗ n r .
Definition 2.9. Let n be an even integer, and let π ∈ P 2 (n). Let f : R n + → C be measurable. The pairing integral of f with respect to π, denoted π f , is defined (when it exists) to be the constant

π f = f (t 1 , . . . , t n ) {i,j}∈π δ(t i -t j ) dt 1 • • • dt n .
We finally introduce the notion of a semicircular family (see e.g. [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Definition 8.15]). Definition 2.10. Let d ≥ 2 be an integer, and let c = {c(i, j) : i, j = 1, ..., d} be a positive definite symmetric matrix. A d-dimensional vector (s 1 , ..., s d ) of random variables in A is said to be a semicircular family with covariance c if for every n ≥ 1 and every (i

1 , ..., i n ) ∈ [d] n ϕ(s i 1 s i 2 • • • s in ) = π∈N C 2 (n) {a,b}∈π c(i a , i b ).
The previous relation implies in particular that, for every i = 1, ..., d, the random variable s i has the S(0, c(i, i)) distribution -see Definition 2.2.

For instance, one can rephrase the defining property of the free Brownian motion S = {S t : t ≥ 0} by saying that, for every

t 1 < t 2 < • • • < t d , the vector (S t 1 , S t 2 -S t 1 , ..., S t d -S t d-1
) is a semicircular family with a diagonal covariance matrix such that c(i, i) = t it i-1 (with t 0 = 0), i = 1, ..., d.

Proof of the main results

A crucial ingredient in the proof of Theorem 1.3 is the following statement, showing that contractions control all important pairing integrals. This is the generalization of Proposition 2.2. in [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF] to our situation. Proposition 3.1. Let d ≥ 2 and n 1 , . . . , n d be some fixed positive integers. Consider, for each i = 1, . . . , d, sequences of mirror-symmetric functions (f

(i) k ) k∈N with f (i) k ∈ L 2 (R n i + ), satisfying: • There is a constant M > 0 such that f (i) k L 2 (R n i + ) ≤ M for all k ∈ N and all i = 1, . . . , d.
• For all i = 1, . . . , d and all p = 1, . . . , n i -1,

lim k→∞ f (i) k p ⌢ f (i) k = 0 in L 2 (R 2n i -2p + ).
Let r ≥ 3, and let π be a connected non-crossing pairing that respects

n i 1 ⊗ • • • ⊗ n ir : π ∈ N C c 2 (n i 1 ⊗ • • • ⊗ n ir ). Then lim k→∞ π f (i 1 ) k ⊗ • • • ⊗ f (ir) k = 0.
Proof. In the same way as in [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF] one sees that without restriction (i.e., up to a cyclic rotation and relabeling of the indices) one can assume that

π f (i 1 ) k ⊗ • • • ⊗ f (ir) k = π ′ (f (i 1 ) k p ⌢ f (i 2 ) k ) ⊗ (f (i 3 ) k ⊗ • • • ⊗ f (ir) k ),
where 0 < 2p < n i 1 + n i 2 and

π ′ ∈ N C c 2 (n i 1 + n i 2 -2p) ⊗ n i 3 ⊗ • • • ⊗ n ir .
Note that 0 < 2p < n i 1 +n i 2 says that f

(i 1 ) k p ⌢ f (i 2 ) k
is not a trivial contraction (trivial means that either nothing or all arguments are contracted); of course, in the case n i 1 = n i 2 it is allowed that p = min(n i 1 , n i 2 ).

By Lemma 2.1. of [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF] we have then

| π f (i 1 ) k ⊗ • • • ⊗ f (ir) k | ≤ f (i 1 ) k p ⌢ f (i 2 ) k L 2 (R n i 1 +n i 2 -2p + ) • f (i 3 ) k L 2 (R n i 3 + ) • • • f (ir) k L 2 (R n ir + ) ≤ f (i 1 ) k p ⌢ f (i 2 ) k L 2 (R n i 1 +n i 2 -2p + ) • M r-2 .
Now we only have to observe that, by also using the mirror symmetry of f

(i 1 ) k and f (i 2 )
k , we have

f (i 1 ) k p ⌢ f (i 2 ) k 2 L 2 (R n i 1 +n i 2 -2p + ) = f (i 1 ) k n i 1 -p ⌢ f (i 1 )
k , f

(i 2 ) k n i 2 -p ⌢ f (i 2 ) k L 2 (R 2p + ) ≤ f (i 1 ) k n i 1 -p ⌢ f (i 1 ) k L 2 (R 2p + ) • f (i 2 ) k n i 2 -p ⌢ f (i 2 ) k L 2 (R 2p + )
. According to our assumption we have, for each i = 1, . . . , d and each q = 1, . . . , n i -1, that

lim k→∞ f (i) k q ⌢ f (i) k = 0 in L 2 (R 2n i -2q + ).
Since now at least one of the two contractions

n i 1 -p ⌢ and n i 2 -p
⌢ is non-trivial, we can choose either q = n i 1p, i = i 1 or q = n i 2p, i = i 2 in the above, and this implies that

lim k→∞ f (i 1 ) k p ⌢ f (i 2 ) k L 2 (R n i 1 +n i 2 -2p + ) = 0,
which gives our claim.

We can now provide a complete proof of Theorem 1.3.

Proof of Theorem 1.3. The equivalence between (2) and (3) follows from [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF]. Clearly, (1) implies (3), so we only have the prove the reverse implication. So let us assume [START_REF] Deya | Convergence of Wigner integrals to the tetilla law[END_REF]. Note that, by Theorem 1.6 of [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF], this is equivalent to the fact that all non-trivial contractions of f (i) k converge to 0; i.e., for each i = 1, . . . , d and each q = 1, . . . , n i -1 we have

(3.1) lim k→∞ f (i) k q ⌢ f (i) k = 0 in L 2 (R 2n i -2q +
).

We will use statement (3) in this form. In order to show (1), we have to show that any moment in the variables I(f

k ), . . . , I(f

(d)
k ) converges, as k → ∞, to the corresponding moment in the semicircular variables s 1 , . . . , s d . So, for r ∈ N and positive integers i 1 , . . . , i r , we consider the moments ϕ I S (f

(i 1 ) k ) • • • I S (f (ir) k ) .
We have to show that they converge, for k → ∞, to the corresponding moment ϕ(s i 1 • • • s ir ). Note that our assumption (1.1) says that lim k→∞ ϕ[I S (f

(i) k )I S (f (j) k )] = c(i, j) = ϕ(s i s j ).
By Proposition 1.38 in [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF] we have ϕ I S (f

(i 1 ) k ) • • • I S (f (ir) k ) = π∈N C 2 (n i 1 ⊗•••⊗n ir ) π f (i 1 ) k ⊗ • • • ⊗ f (ir) k . By Remark 1.33 in [4], any π ∈ N C 2 (n i 1 ⊗ • • • ⊗ n ir ) can be uniquely decomposed into a disjoint union of connected pairings π = π 1 ⊔ • • • ⊔ π m with π q ∈ N C c 2 ( j∈Iq n i j )
, where {1, . . . , r} = I 1 ⊔ • • • ⊔ I m is a partition of the index set {1, . . . , r}. The above integral with respect to π factors then accordingly into

π f (i 1 ) k ⊗ • • • ⊗ f (ir) k = m q=1 πq j∈Iq f (i j ) k .
Consider now one of those factors, corresponding to π q . Since π q must respect j∈Iq n i j , the number r q := #I q must be strictly greater than 1. On the other hand, if r q ≥ 3, then, from (3.1) and Proposition 3.1, it follows that the corresponding pairing integral πq j∈Iq f (i j ) k converges to 0 in L 2 . Thus, in the limit, only those π make a contribution, for which all r q are equal to 2, i.e., where each of the π q in the decomposition of π corresponds to a complete contraction between two of the appearing functions. Let N C 2 2 (n i 1 ⊗ • • • ⊗ n ir ) denote the set of those pairings π. So we get lim k→∞ ϕ I(f

(i 1 ) k ) • • • I(f (ir) k ) = π∈N C 2 2 (n i 1 ⊗•••⊗n ir ) lim k→∞ π f (i 1 ) k ⊗ • • • ⊗ f (ir) k ,
We continue as in [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF]: each π ∈ N C 2 2 (n i 1 ⊗• • •⊗n ir ) is in bijection with a noncrossing pairing σ ∈ N C 2 (r). The contribution of such a π is the product of the complete contractions for each pair of the corresponding σ ∈ N C 2 (r); but the complete contraction is just the L 2 inner product between the paired functions, i.e., lim k→∞ ϕ I S (f

(i 1 ) k ) • • • I S (f (ir) k ) = σ∈N C 2 (r) {s,t}∈σ c(i s , i t ).
This is exactly the moment ϕ(s i 1 • • • s ir ) of a semicircular family (s 1 , . . . , s d ) with covariance matrix c, and the proof is concluded.

We conclude this paper with the proof of Theorem 1.6. for every n, m ≥ 0, and every f ∈ L 2 (R n + ), g ∈ L 2 (R m + ). For point (2), we observe first that the case d = 1 is already known, as it corresponds to [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF]Theorem 1.8]. Consider now the case d ≥ 2. Let us suppose that I S (f

  Partitions. A partition of [n] = {1, 2, . . . , n} is (as the name suggests) a collection of mutually disjoint nonempty subsets B 1 , . . . , B r of [n] such that B 1 ⊔ • • • ⊔ B r = [n]. The subsets are called the blocks of the partition. By convention we order the blocks by their least elements;

Proof of Theorem 1 . 6 .

 16 Point (1) is a simple consequence of the Wigner isometry (3.2) (since each f(i) k is fully symmetric, f (i)k is in particular mirrorsymmetric), together with the classical Wiener isometry which states that(3.2) E[I W (f )I W (g)] = 1 n=m × n! f, g L 2 (R n + )

→

  (s 1 , . . . , s d ). In particular, I S (f(i) k ) law → s i for all i = 1, . . . , d. By [4, Theorem 1.8] (case d = 1), this implies that I W (f (i) k ) law → (n i )!N i . Sincethe asymptotic relations in (1) are verified, Theorem 1.1(B) leads then to I W (f (1) k ), . . . , I W (f (d) k ) law → (n i )!N 1 , . . . , (n d )!N d , which is the desired conclusion. The converse implication follows exactly the same lines,and the proof is concluded.