Philippe Le Goffe

Should we promote grassland ? A microeconomic analysis gsorr ResùftffÈto=*

Keywords: grassland, multifonctionality, policy, extemality, transaction costs, jointness

Unfavourable or no incentives at all make the farmers use insufficient grassland areas, from the society's and economic efficiency's point of view. On the one hand the CAP premium encourage to exploit more crops and less grasslands than what real profit would imply. On the other hand, the farmer who seeks to maximise his profit does not take into account the effects extemal to the market due to the lack of incentives. However the market could contribute to the provision of grasslands the society needs, because milk and meat produced by grass-feeding have specific characteristics valued by the consumer. Nevertheless, the public good characteristics of grasslands make market failures remain and public policy become necessary. The conception of efficient policies for grasslands'multifunctionality requires the survey of transaction costs and jointness between breeding and grassland. The different kinds of grassland premium probably allow to make environmental benefits without increasing excessively the transaction costs in comparison to price support.

Introduction

The OECD (2000) defines the mutifunctionality as a distinctive trait of agriculture, in other words the capacity to produce both base products (food) and other products. When these other products are characteized as public good, the markets do not do well and the supply may then be inadequate or insuffrcient. Grass feeding systems are typically multifunctional, which fits the OECD's definition since grasslands offer not only meat and milk productions but also several collective benefits. As a matter of fact, grasslands contribute to a high degree to the struggle against erosion and to the regularizing of water regime or flow ; to the purification of fertilizers and pesticides, and to biodiversity. They, at last, have aesthetic roles (landscape) and recreational firnctions as far as they provide public access that other agricultural uses do not allow.

However, although grasslands have a real social interest, a steady decrease of surfaces in permanent grassland has been noticed in France for 30 years, the equivalent of Io a year . At the same time, since 1992,the European Union and the state members have been trying to set up policies aiming at promoting grassland, in order to meet the social demand. Among those measures grassland premium and some measures for the Farming Telritorial Contracts (FTC) in France can be quoted. Bearing this context in mind, this article's purpose is to present the economic mechanisms which explain why there are not enough grasslands from society's point of view. Then we will examine how the market and the public policies might efficiently provide the grasslands the society needs.

The first part deals with the microeconomic modelisation of the producer's behaviour. One can see how the presence or the lack of signals encourages the producer to deviate from the supply of grasslands necessary for society. The second part tries to apply to the case of grasslands the recent studies on the policies for multifunctionality and their interaction whith agricultural policies.

1. Producer's behaviour, market and agricultural policies failures

The microeconomic modelisation of the producer's behaviour provides aî analyical framework which helps to explain how the producer divides his farmlands between crops and grasslands (see [START_REF] Guyomard | La nouvelle instrumentation de la politique agricole conrnrune[END_REF]. Let us take the example of a farmer who produces agricultural goods from grasslands and crops. Let us make the classical hypothesis in microeconomics that the producer, a selÊinterested and rational individual, wants to maximise his private profit knowing that his total agricultural area is constrained (see appendix). The programme's resolution of the producer leads to the equality of marginal profits taken from an extra hectar of grassland or crop. The marginal profit of an extra hectare is the extra value of milk or meat production diminished by variable costs per hectar of grassland or crop (according to accounting tems, this is a gross margin).

Figure I

Because of the hypothesis of decreasing yields, true to what is being observed in agriculture, the marginal profits of grasslands and crops, shown in figure l, are decreasing functions. We also have to take into account the incentives provided by the Common Agricultural Policy (CAP). The CAP crop premium increases the marginal profit of crops, so that the equality of marginal profits takes place for a grassland area of x,n. The producer will then choose a grassland area equivalent to x-.

However since agriculture is a multifunctional activity, other products and more particularly public goods are being produced in addition of base products (vegetal and animal ones). Public goods are different from private goods as far as one cannot prevent people from enjoying them (the landscape example). This impossibility of exclusion resulting from the collective consumption is responsible for the lack of market for public goods whether they are goods or bads. As a consequence there are interactions taking place between agents with no monetary compensation. Thus farmers do not secure spontaneously the victims of agricultural pollution. At the same time the collective services provided by the farmers cannot be paid by consumers who benefit them. These effects that the market cannot control are called as extemalities or external effects. The lack of market incentives is at the heart of an excessive pollution and of insufficient collective services. The concern for maximising the collective value for society leads to making these extemal effects paid by the agents, that is to say by internalising them.

In order to identify this optimum solution in a collective way, we represented the extemal effects in figure 1. Crop farming generally has extemal negative effects : erosion, pollution linked to the bad use of fertilizers and phytosanitary products, even to landscape damages (Poux, 2000). These environmental damages decrease the real marginal profit of crops (figure 1). On the contrary, as indicated above, grasslands provide collective services whose value must be added to the marginal profit. The consequence is that the equality of cor:rected marginal profits for the external effects (and for CAP distortion) is now taking place in x*, which corresponds to the optimum area of grasslands for society. The CAP distortion and the market failures result in the farmers using insufficient areas, the consequence being a social loss represented by the shaded area at figure 1. The social loss comes from the differential premium which alters the profitability of crops, from the damages inflicted to the environment and the loss of income linked to the lack of collective services not provided by grasslands. In the second part, we shall see how the market and the public action should allow to go toward an optimal solution.

2.The optimal supply of grasslands by the market and the public policies

Context and challenges

The problem of promoting grasslands in cattle systems is a specific case of the more general debate about the multifunctionality of agriculture. In european countries, the authorities have started to reply to the citizens'request for a multifirnctional agriculture (agri-environmental measures of the CAP, FTC in France, etc...). The policies for multifunctionality are obviously linked to classical agricultural policies acting on the markets and the prices, whose reform is at stake in the negociations with the OMC. The subsidies to the agricultural production and the frontiers'protection are being blamed for creating distortion of competition on the markets and the exchanges, by stimulating the agricultural production and the use of input (one can talk of coupled measures when the support directs the production and the markets). While a pressure aiming at decreasing agricultural subsidies is real, the trading partners on the intemational scene fear that the multifunctionality may be used to maintain the distortion of competition which favour agricultures traditionally highly subsidized (Europe, Japan).

In this context how can we conceive policies for multifunctionality which are economically justified and how do the latter interact with classical agricultural policies ? The OECD has tried to answer these questions, especially in a document entitled < Multifunctionality : the building up of an analfiical framework ) (OECD, 2000), and at a workshop which took place in July 200L

These works have enlightened three questions which arc at the heart of the conception of efficient policies for multifunctionality :

the question of the market failures that we dealt with in the first part and which justifies public action, the question of the jointness of production between other products and base products which is essential in the cost of production of public goods (concept of range economics), the question conceming transaction costs observed during the setting up of public policies or during trading transactions.

Economically speaking, it is legitimate to look for supplying the public goods which society needs, at the lowest possible cost including transaction costs and provoking the least possible distortions in the markets and the exchanges. Following this we will endeavour to explain those notions and will apply them to the particular case of grasslands.

Joint production and transaction costs

The joint production of base products and other products is due to the complementarities of the production, which we qualify as technical interactions, or is due to the fact that the multiple products are obtained with only one input (non affectable input). The technical interactions are for example the source of agricultural pollutions. In the case of cattle grazing, grassland is the non affectable input which allows to produce at the same time the meat or the milk and the public goods associated with grasslands. However, in agriculture the joint productions are rarely produced on a steady proportion, because there are numerous production systems which use different technologies and agricultural practices. In other words the jointness is not always strong and the dissociation between the joint productions is often possible. This explains that cattle breeding can be practised with or without grasslands, and with environmental results which differ whether grasslands are permanent or cultivated (this has an impact on biodiversity and erosion), or according to the level of fertilization and the use of chemicals for plant protection. Even with organic agriculture, an example of strong jointness between base products and public goods, the environmental benefit compared to conventional agriculture is better for crop protection and nitrogen balance than for biodiversity [START_REF] Rainelli | Rentabilité comparée de l'agriculture biologique et de l'agriculture conventionnelle : incidence du soutien public sur les grandes cultures en France[END_REF].

A serious issue to be considered is if dissociation is costly or not. As a matter of fact meat and milk can be produced without grass and grassland areas can be tended by non agricultural firms. But in this case, the cost will be higher because the rural firm will have to be paid for a service which would have been provided at a low cost by the farmers. The jointness in production is responsible for range economics, which means that the cost of joint production is lower than the one of separated production of multiple products. Grass feeding systems thus creates range economics, which makes it, generally speaking, the cheapest solution to provide public goods associated with grasslands. In other agricultural situations, the dissociation is possible without cost and the non agricultural providing of public goods can then be taken into account (example: the tending of hedges, banks, walls...). This can also affect the tending of grasslands, in the areas of specialised agriculture where grass-feeding animals disappeared, and where its reintroduction is extremely costly. But in this case, the quality of public goods has also to be taken into account : does the consumer enhance in the same way the aesthetics, ecological and recreational qualities of grasslands whether they are grazed or only cut ?

The definition of transaction costs offered by Coase (1960), in the case of market transactions can be enlarged to different kinds of transactions, whether they are being done on the market or by a government, whether they imply a financial transfer or not [START_REF] Challen | Non-govemment approaches to the provision of non-commodity agricultural outputs : a transaction-cost perspective[END_REF]. These are all the costs associated to the necessary operations to the making of a transaction : prospection, agents'information, negotiation, making of a contract, its control and its application... These costs of running the economy have been compared to frictions in physical systems ; as too much friction prevents movement, the more expensive the costs of transactions are, the more they prevent the market from working well when they are higher than expected benefits of market transactions.

Although they are the same type of costs, it is nevertheless convenient to make a difference between the transaction costs observed on the market and the ones which are linked to the setting up of public policies (OECD, 2001). In the case of environmental goods provided by agriculture, Challen (2001) offered a typology for the transaction costs which may be an obstacle to the fi.rnctioning of the market. Notably, there are transaction costs linked to the characteristics of credence goods and of public goods. The environmental characteristics of agricultural products are as a maiher of fact credence goods, whose quality cannot be checked neither at the purchase nor at the consumption, unless one starts checking the steps which would be costly for the consumer. This assymmetry of information is the source of high transaction costs. The characteristics of public goods make it very difficult and costly to exclude consumers from consuming those goods. These difficulties favour a type of behaviour called << free-rider > (taking advantage of a good without paying for it ), which is antinomic to the market. So the costs of exclusion are equally serious transaction costs.

2.3Interest and limits of the market

If the transaction costs observed on the market are inferior to the benefits potentially drawn from the transactions, then the spontaneous providing of other products by the market is possible. This is for example what we can frequently notice for the practise of fishing and hunting, because the exclusion costs are low in this case and they are not credence goods (this market management is less frequent in France than in other european countries for institutional reasons). When the non-govemmental providing of other products is possible, it is interesting because it is less expensive and often allows a better adjustment of the supply to the demand than the public one ; however, by excluding consumers it may create a problem of equity. When there is no market, one can seek to decrease the transaction costs in order to allow the non-government providing. [START_REF] Challen | Non-govemment approaches to the provision of non-commodity agricultural outputs : a transaction-cost perspective[END_REF] shows how the eco-labelling of agricultural products reduces the transaction costs linked to the credence goods because the assymetry of information is being reduced.

That is why the State has to promote the ecolabelling and the systems of quality warranty (systems of environmental management), in the initial phase of their development where profit is not guaranteed. On the other hand, decreasing the exclusion costs is difficult when the other products arc rcal public goods, which is very often the case of environmental characteristics of agricultural goods. This incentive to the free-rider may explain the low market share of green products (Ifen, 1994). Surueys reported by Ifen (1993) show that consumers motivated to buy organic products do it for health reasons (private attributes) rather than for environmental concems (public attributes) ; here however, by joining specific private attributes and public attibutes, this process of production allows the market supply of base products and other products.

In a survey presented in this congress, [START_REF] Coulon | Influence of forage feeding on the composition and sensory properties of meat and dairy products : bases for a terroir effect[END_REF] show that grass-feeding influences the meat and milk characteristics (nutritional, technological, hygienic and sensory). These constitutive characteristics of the quality of the products are private attributes for the consumer, which are going to merchandise well if the consumer looks for them. In this case the strong jointness between the public attributes and the private onesthose characteristics of the products being specific of grass feeding -implies that the market is going to provide the public goods associated to grasslands. Some of these characteristics have been sought for very long in the AOC framework. Thus in the Comté (pressed cooked cheese) Protected Denomination of Origin (PDO), it is essentially technological and sensorial arguments which led to impose constraints on grass feeding (norm conceming the minimal grassland area, silage and other fermented food are forbidden, forages must come from Comté's area, etc...).

Other cheese PDO are right now debating on silage ban, given the sanitary risks of contamination by pathogen bacteria (listeria mainly).

When the technological link is not so obvious, the market will probably be failing and a labeling based only upon public goodhere grasslands -will not have any success ; this explains why, as far as we know, a specific quality sign < grass feeding > does not exist right now on the market. On the other hand, the public goods associated to grasslands takes part in the collective image of a sign of quality, evenmore so when the region of production is known (PDO, Protected Geographic Indication). However, as we saw in the first paft, the farmer is not economically incited to resort to grasslands as much as it should be expected. Although the non-production of public good is collectively detrimental, the individual strategy of the agents leads them not to cooperate for this production (behaviour qualified in economy as << the prisoner's dilemma )). A collective solution to this problem consists in introducing constraints conceming grasslands in the requirements associated to quality signs, which results in creating the jointness between private and public attributes of agricultural products.

As an example we are pleased to notice the reenforcing of grass feeding in the requirements of the different French milk or meat PDO (Coulon, personal communication). This policy could be applied to other quality signs.

In this case of technological or statutory jointness between the private and public attributes of agricultural products, the market fits the supply of private attributes, but not necessarily the one conceming public attributes obtained in addition. Market failures will remain linked to the fact that the amateurs of the services provided by grasslands are not necessarily the consumers of quality products produced by grass-feeding cattle, and that the consumption of those goods is not in the same ratio as their production.

Can the market of rural tourism contribute to giving the grasslands surfaces wanted by the society ? According to the data coming from the Britanny region (western part of France), Le [START_REF] Goffe | Hedonic pricing of agriculture and forestry extemalities[END_REF] showed that the renting price of rural cottages from a rural zone increased with the importance of permanent grasslands in the agricultural lands of the zone. The problem lies in the factthatthe consumer is sensitive to the environmental attributes of a large rural areaamong others are grasslands -and not only to the ones of the lands which are in the immediate surroundings of the rural cottage (which belong sometimes to the cottage's landlord ). As the landscape of the rural area is a public good, the farmers who are incited to replace grasslands by crops, do not compensate the cottages'landlords for the loss of profit which they suffer from (case of externality). In the same way, the consumers of tourism do not have the possibility to pay the grasslands producers of the area. Even if we imagine that the farmers are also the owners of the rural cottages, the individual strategy of the agents implied by the << prisoner's dilemma > would lead to areas of grasslands too little to maximise the income of the arca, because the agents do not have any interest in cooperating (see Mahé and Ortalo- Magné, 2001). The characteristic of public good linked to landscape (the lack of property rights) is the source of too high exclusion costs for the market to be functioning (but transaction costs linked to credence goods could also be present since the environmental quality is not necessarily known to the consumer). This situation, as the restrictions mentioned in the former case, justiS to resort to public policies.

2.4 Should public policies be targeted or coupled ?

Here again, the type of jointness and the governmental transaction costs are the central elements which largely condition the type of public policy to set up and interaction with the classical agricultural policies (OECD, 2000). If the base products and the other products are not joined, the policies for multifunctionality will be targeted on the public goods and independant from the agricultural policies. In this case, the lack of range economics may make the non-agricultural providing of public goods competitive (case of hedges, banks and walls, for their market fi.rnction has almost disappeared, except for some alternative production systems). [START_REF] Vatn | Transaction costs and multifunctionality[END_REF] studied the opposite case where there is a total link between the multiple products, and where the provision of other products is insufficient (market failure at worldmarket price). Vatn (2001) shows that we would get an apparently similar result, either by supporting the price of base products (coupled policy) until one obtains the desired level of other products, or by targeting the support on the provision of public goods (through agro- environmental contracts), since the multiple products are linked. However, the coupled policy will be systematically prefened because it allows to save the transaction costs linked to the agro-environmental contracts. The same arguments would lead to tax the price of base products, in the case where the other products were negative extemalities (for example pollution).

Figure 2 As we saw it, the reality of agriculture often conesponds to the intermediary case where the jointness is partial. Here the choice between coupling and targeting depends on the trade-off between the public benefits and the transaction costs of the targeting. On the figure 2 taken from Vatn (2001), one can see that the finer and finer targeting provides decreasing marginal benefits and creates increasing marginal transaction costs. An economic optimum is reached when marginal benefits equal marginal transaction costs of targeting ; beyond it, the targeting is inefficient because the extra benefits are overshot by the higher transaction costs.

We saw that grasslands fit with this situation of defective jointness. In order to clear up the choice of policies for grasslands'multifunctionality, it is important to better understand the jointness between cattle-breeding and grasslands : are the quantity, the quality and the grasslands'localisation satis$ing ? What is the impact of milk or meat price support on the use of grasslands ? Would its suppression lead to the neglect of grasslands surfaces ? What are the negative externalities implied by the price support ? IVhat is the loss of environmental benefits in comparison to a more precise targeting ? At the same time, one needs figures on the transaction costs associated to the agro-environmental contracts, which have been scarcely studied until now. A policy supporting the prices of milk or meat could be economically justified because of the low transaction costs it creates, although it produces less public services than atargeted policy (for it is less accurate), and more negative externalities.

One can nevertheless think that a support based on the non affectable input, that is to say a grassland premium, would permit to produce more public benefits linked to grasslands and less negative extemalities, without increasing in an excessive way the transaction costs (this depends on the contracts'complexities). In all the cases, no economic argument justifies differential premium favouring crops, as far as they produce less public goods than grasslands.

Table 1 In France, farmers have access to a wide range of measures aiming at promoting grassland (table 1). The premium to the maintaining of extensive breeding systems (PMEBS, the former grassland premium) is the more general one : the requirements only concern the minimal proportion of grassland surface, the nitrogen fertilization and the stocking rate. The measures included in the FTC are better targeted : patch of land with high plant-care risk, grassed stripes near the waters. The competition with the CAP crop premium leads to paying high- price grassland measures, whose extension is necessarily limited and makes the least subsidized measures little attractive, mainly in the intensive agricultural areas (Le Goffe and [START_REF] Le Goffe | Les CTE en Bretagne : des principes économiques aux réalités[END_REF]. Once these distortions are suppressed, the analysis of environmental benefits of targeting and the transaction costs would allow, here again, to choose between general grassland premium and targeted contracts. The choices would probably be different according to the regions, for the targeting benefits depend on the regional context. In this perspective, it would be interesting to test even less constraining measures than the PMEBS (no fertilization or stocking requirements), as a simple premium to grassland areas (like the CAP crop premium), or a premium to permanent grassland, which would have the advantage of diminishing the checking costs while targeting the most interesting grasslands (biodiversity, erosion,etc. ..).

Conclusion

Unfavourable or no incentives at all make the farmers use insufficient grassland areas, from the society's and economic efficiency's point of view. On the one hand the CAP premium encourage to exploit more crops and less grasslands than what real profit would imply. On the other hand, the farmer who seeks to maximise his profit does not take into account the effects which get out of the market's hand, because of the lack of incentives. The result is an excess of environmental damages due to the crops and a shortage of collective services offered by thegrasslands, that is to say a loss of collective value. The intemalisation of these extemal effects would correct the rotation in favour of grasslands.

The market may contribute to provide the grasslands'areas society needs, because milk and meat produced by grass-feeding have particular characteristics which can be enhanced toward the consumers, with low enough transaction costs. However, the characteristics of public good of grasslands may make market failures last, such as the ones linked to the < free-rider > syndrome or to the << prisoner's dilemma >.

Public policies for grasslands'multifunctionality are then necessary. The economic justification of the choice between coupled or targeted policies depends upon the trade-off between the environmental benefits and the transaction costs of the targeting. The different kinds of grassland premium probably allow to make environmental benefits without increasing in a high proportion the transaction costs in comparison to the price support. On the other hand, the differential premium favouring crops have no economic justifications. The conception of efficient policies for grasslands'multifunctionality make it necessary to start economic research on the jointness between breeding and grasslands and on the measurement oftransaction costs.

Table 1 .

 1 Measures aiming at promoting grassland (France)MeasureConversion or preservation Subsidy (per hectare per year) Conversion of arable lands whith high planlcare risk into Trade-off between crops and grasslands : private and social optima.Figure2. Trade-off between benefits and transaction costs of targeting(from Vatn, 2001).

		Conversion	375 € (crops)
			305 € (cultivated grasslands)
	grasslands
	Set up of grassed stripes along	Conversion	375 C (crops)
	rivers
	Adoption of a production	Conversion or preservation	131 C (for the whole farm
	system mainly based on		area)
	grasslands and low inputs
	Extensive grassland	Preservation	76 € (concerned area)
	management
	Premium to the maintening of	Preservation	46 € (concerned area)
	extensive breeding systems

t2 Figure 1. costs targeting

Appendix : Maximisation program of farmer's private profït fi(xù is the production function which describes the production of an agricultural product, which plice is p1, from grassland arcaxt.

Similarly,f2(x) is the production function which describes the production of an agricultural product, which pice is p2, from crop àtêàx2.

Xis the total agricultural area available to the farm.

w 1 arrd w2 respectively design operating costs per hectare of grasslands and crops and FC the farm's fixed costs. The producer's profit maximisation program can be written as follows : max lT = (p r.fr(", ) -w rx,) + (p, f , (. It follows the equality between marginal profits from grasslands and crops


M Bergère

B Eynard

Some properties of angular integrals

We find new representations for Itzykson-Zuber like angular integrals for arbitrary β, in particular for the orthogonal group O(n), the unitary group U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formula on n. The same methods gives also the Shatashvili's type moments. Finally we prove that, in agreement with Brezin and Hikami's observation, the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables (x i -x j )(y i -y j ).

Introduction

What we call angular integral [START_REF] Mehta | Random Matrices[END_REF] is an integral over a compact Lie group G β,n :

G 1/2,n = O(n) , G 1,n = U(n) , G 2,n = Sp(2n) (1-1)
of the form:

I β,n (X, Y ) = G β,n dO e Tr XOY O -1 (1-2)
where X and Y are two given matrices, and dO is the Haar invariant measure on the group. We shall also extend I β,n to arbitrary β (Notice that our β is half the one most commonly used in matrix models, for instance we have β = 1 in the unitary case).

In this paper we are going to consider the case where X and Y are diagonal matrices, however, let us first recall the Harish-Chandra case.

Harish-Chandra case

In the case where X and Y are in the Lie algebra of the group [START_REF] Knapp | Representation theory of semisimple groups[END_REF] (i.e. real antisymmetric in the O(n) case, anti-hermitian in the U(n) case, and quaternion-anti-selfdual in the Sp(2n) case), the angular integral can be computed with Weyl-character formula, and is given by the famous Harish-Chandra formula [10] (which is also a special case of the Duistermaat-Heckman localization [START_REF] Duistermaat | [END_REF]):

(X, Y ) ∈ Lie algebra ⇒ dO e Tr XOY O -1 = C w∈Weyl e Tr XYw ∆ β (X) ∆ β (Y w ) (1-3)
where C is a normalization constant, w runs over elements of the Weyl group, and the generalized Vandermonde determinant ∆ β (X) is the product of scalar products of positive roots with X (see [10,[START_REF] Ferrer | Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups[END_REF][START_REF] Knapp | Representation theory of semisimple groups[END_REF] for details).

Diagonal case

However, for applications to many physics problems [START_REF] Dyson | Statistical Theory of the Energy Levels of Complex Systems. I[END_REF][START_REF] Mehta | Random Matrices[END_REF], it would be more interesting to have X and Y in other representations, and in particular X and Y diagonal matrices.

Since a antihermitian matrix is, up to a multiplication by i, a hermitian matrix, and since every hermitian matrix can be diagonalized with a unitary conjugation, for the unitary group, the Harish-Chandra formula applies as well to the case where X and Y are diagonal, this is known as Itzykson-Zuber formula [18]:

X = diag(x 1 , . . . , x n ) Y = diag(y 1 , . . . , y n ) ⇒ U (n)
dU e Tr XU Y U -1 = C n det e x i y j ∆(X) ∆(Y ) [START_REF] Abramowitz | Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables -1974[END_REF][START_REF] Baker | The Calogero-Sutherland Model and Generalized Classical Polynomials[END_REF][START_REF] Brezin | An extension of the Harish-ChandraItzykson-Zuber integral[END_REF][START_REF] Calogero | Ground State of a One-Dimensional N-Body System[END_REF] where ∆(X) = ∆ 1 (X) = i>j (x i -x j ) is the usual Vandermonde determinant.

For the other groups, computing angular integrals has remained an important challenge in mathematical physics for a rather long time. Many progresses and formulae have been found, however, a formula as compact and convenient as Harish-Chandra is still missing. And in particular a formula which would allow to compute multiple matrix integrals, generalizing the method of Mehta [START_REF] Mehta | A method of integration over matrix variables[END_REF] is still missing.

Calogero Hamiltonian

It is known that, in the diagonal case, I β,n satisfies the Calogero-Moser equation [START_REF] Calogero | Ground State of a One-Dimensional N-Body System[END_REF], i.e. is an eigenfunction of the Calogero hamiltonian:

H Calogero .I β,n = ( i y 2 i ) I β,n (1-5)
H Calogero = i ∂ 2 ∂x 2 i + β j =i 1 x i -x j ( ∂ ∂x i - ∂ ∂x j ) (1-6)
Many approaches towards computing angular integrals have used that differential equation. A basis of eigenfunctions of the Calogero hamiltonian is the Hi-Jack polynomials [START_REF] Calogero | Ground State of a One-Dimensional N-Body System[END_REF][START_REF] Baker | The Calogero-Sutherland Model and Generalized Classical Polynomials[END_REF][START_REF] Desrosiers | Duality in random matrix ensembles for all Beta[END_REF][START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF].

In particular remarkable progress in the computation of I β,n was achieved recently by Brezin and Hikami [START_REF] Brezin | An extension of the Harish-ChandraItzykson-Zuber integral[END_REF]. By decomposing I β,n on the suitable basis of Zonal polynomials, they were able to find a recursive algorithm to compute the terms in some power series expansion of I β,n , and they obtained a remarkable structure. In particular they observed that the power series reduces to a polynomial when β ∈ N.

Morozov and Shatashvili's formulae

Another important question for physical applications, is not only to compute the angular integral (the partition function in statistical physics language), but also all its moments, for instance:

M i,j = G β,n dO e Tr XOY O -1 ||O i,j || 2 (1-7)
and more generally for any indices i 1 , . . . , i 2k , j 1 , . . . , j 2k :

G β,n dO e Tr XOY O -1 O i 1 ,i 2 O i 3 ,i 4 . . . O i 2k-1 ,i 2k O -1 j 1 ,j 2 O -1 j 3 ,j 4 . . . O -1 j 2k-1 ,j 2k
(1-8)

In the U(n) case β = 1, Morozov [28] found a beautiful formula for M i,j , and Shatashvili [30] found a more general formula for any moments of type 1-8 using the action-angle variables of Gelfand-Tseytlin corresponding to the integrable structure of this integral.

For β = 1/2, 1, 2, in the Harish-Chandra case where X and Y are in the Lie algebra, a formula for all possible moments was also derived in [START_REF] Ferrer | Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups[END_REF], generalizing Morozov's [START_REF] Eynard | A short note about Morozovs formula[END_REF]13].

In this article we shall propose new formulae for M i,j in the diagonal case for arbitrary β, and our method can also be generalized to all moments.

Outline of the article

• Section 1 is an introduction, and we present a summary of the main results of this article.

• In section ?? we setup the notations, and we review some known examples.

• In section 4 we show how to transform the angular integral with a Haar measure into a flat Lebesgue measure integral on a hyperplane. From it, we deduce a recursion formula, as well as a duality formula (the angular integral is an eigenfunction of kernel which is the Cauchy determinant to the power β).

• In section 5, we discuss the moments of the angular integral. We show that moments can be obtained also with Lebesgue measure integrals, and we show that they satisfy linear Dunkl-like equations. This can be used as a way to recover Calogero equation for the angular integral.

• In section 6, we rewrite the angular integral as a symmetric sum of exponentials with polynomial prefactors. Those polynomials are called principal terms, and can be computed recursively. In particular, we prove the conjecture of Brezin and Hikami [START_REF] Brezin | An extension of the Harish-ChandraItzykson-Zuber integral[END_REF] that the principal terms are polynomials in some reduced variables (x i -x j )(y i -y j ).

• In section 6.2, we prove a formula for n = 3 in terms of Bessel polynomials, and we propose a conjecture formula for arbitrary β and arbitrary n.

• In section 6.4, we focus on the symplectic case β = 2, for which we can improve the recursion formula.

• Section 7 is the conclusion.

• Appendices contain useful lemmas, and proofs of the most technical theorems.

Summary of the main results presented in this article

• We rewrite the angular integral with the Haar measure on the Lie group G β,n , as a flat Lebesgue measure integral on its Lie algebra (notations are explained in section 4):

I β,n (X; Y ) = dO e Tr XOY O -1 = dS e Tr S n k=1 det(S -y k X) β (1-9)
as well as its moments:

M i,j = dO ||O i,j || 2 e Tr XOY O -1 = β dS e Tr S n k=1 det(S-y k X) β ((S -y j X) -1 ) i,i (1-10)
• We show that the M i,j 's satisfy a linear functional equation (very similar to Dunkl operators):

∀i, j, ∂M i,j ∂x i + β k =i M i,j -M k,j x i -x k = M i,j y j (1-11)
which implies the Calogero equation for

I β,n = i M i,j = j M i,j : i ∂ 2 I β,n ∂x 2 i + β j =i 1 x i -x j ( ∂I β,n ∂x i - ∂I β,n ∂x j ) = ( i y 2 i ) I β,n (1-12)
Moreover, the integral of eq. (1-10), is a solution of the linear functional equation eq. [START_REF] Abramowitz | Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables -1974[END_REF][START_REF] Baker | The Calogero-Sutherland Model and Generalized Classical Polynomials[END_REF][START_REF] Brezin | An extension of the Harish-ChandraItzykson-Zuber integral[END_REF][START_REF] Calogero | Ground State of a One-Dimensional N-Body System[END_REF][START_REF] Carlitz | A Note on the Bessel Polynomials[END_REF][START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF][START_REF] Desrosiers | Duality in random matrix ensembles for all Beta[END_REF][START_REF] Duistermaat | [END_REF][START_REF] Dyson | Statistical Theory of the Energy Levels of Complex Systems. I[END_REF](10)[START_REF] Eynard | A short note about Morozovs formula[END_REF] for any choice of integration domain (as long as there is no boundary term when one integrates by parts). We thus have a large set of solutions of the linear equation, and also of Calogero equation.

• We deduce a duality formula:

I β,n (X; Y ) = det(X) 1-β dλ 1 . . . dλ n ∆(Λ) 2β I β,n (X, Λ) n k=1 n j=1 (λ j -y k ) β (1-13)
and a recursion formula:

I β,n (X; Y ) = e xn n i=1 y i n-1 i=1 (x i -xn) 2β-1 dλ 1 , . . . dλ n-1 I β,n-1 (X n-1 ,Λ) ∆(Λ) 2β e -xn i λ i n k=1 n-1 i=1 (λ i -y k ) β (1-14)
similar to that of [15,16].

• For β ∈ N, the solution of the recursion can be written in terms of principal terms:

I β,n (X, Y ) = σ e n i=1 x i y σ(i) ∆(X) 2β ∆(Y σ ) 2β Îβ,n (X, Y σ ) (1-15)
where σ is the sum over all permutations.

The recursion relation eq. (1-14) can be rewritten as a recursion for the principal terms Îβ,n (X, Y ):

Îβ,n (X n ; Y n ) = ∆(Y n ) 2β (β -1)! n-1 n-1 i=1 x i,n ∂ ∂a i β-1 Îβ,n-1 (X n-1 , a) e i x i,n (a i -y i ) n k=1 n-1 i=1, =k (y k -a i ) β a i =y i (1-16)
• For general n and β integer, we prove the conjecture of Brezin and Hikami [START_REF] Brezin | An extension of the Harish-ChandraItzykson-Zuber integral[END_REF], that the principal term Îβ,n (X, Y ) is a symmetric polynomial of degree β in the τ i,j variables,

τ i,j = - (x i -x j )(y i -y j ) 2 (1-17)
• In the case n = 3 we find this polynomial explicitly for any β (for β integer the sum is finite):

I β,3 ∝ e x 1 y 1 +x 2 y 2 +x 3 y 3 (∆(x)∆(y)) β ∞ k=0 Γ(β -k) 2 6k k! Γ(β + k) i<j Y (k) β-1 ( 1 τ ij ) + sym (1-18)
where Y m is the m th Bessel polynomial, i.e. the modified Bessel function of the second kind (see definition of Y β-1 below in eq. (3-4)).

• In the case β = 2 (i.e. symplectic group Sp(2n)), the recursion relation for the principal term can be written:

Î2,n = n-1 i=1 (x i -x n ) (y i -y n ) 2 n-1 i=1 x i -x n -n k=1, =i 2
y i -y k + ∂ ∂a i Î2,n-1 (X n-1 , a) a i =y i = ∆(X n ) 2 ∆(Y n ) 2 det X n-1 -xn-2 Y n-1 -yn +B+∂ Y det(Xn-xn) Î2,n-1 (X n-1 ;Y n-1 ) ∆(X n-1 ) 2 ∆(Y n-1 ) 2 (1-19)
and B is the antisymmetric matrix B i,j = √ 2 y i -y j , B i,i = 0, and ∂ Y = diag(∂ y 1 , . . . , ∂ y n-1 ). In section 6.4 we propose an operator formalism to compute it, and we propose a conjecture formula in terms of decomposition into triangles.

Definitions and examples 3 secdefex

Notations for angular integrals

Let X and Y be two diagonal matrices of size n:

X = diag(x 1 , . . . , x n ) , Y = diag(y 1 , . . . , y n ) (3-1)
We define the angular integral:

I β,n (x 1 , . . . , x n ; y 1 , . . . , y n ) = G β,n dO e Tr XOY O -1 (3-2)
where G β,n denotes one of the Lie groups:

G 1/2,n = O(n) , G 1,n = U(n) , G 2,n = Sp(2n) (3-3)
and dO is the invariant Haar measure on the corresponding compact Lie group. We will later extend those notions to arbitrary values of β.

Bessel polynomials

For further use, we need to introduce some Bessel functions [START_REF] Abramowitz | Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables -1974[END_REF][START_REF] Krall | A New Class of Orthogonal Polynomials: The Bessel Polynomials[END_REF]31,32,[START_REF] Carlitz | A Note on the Bessel Polynomials[END_REF]. Those special functions are going to play a major role throughough this article.

The Bessel polynomials (see [START_REF] Krall | A New Class of Orthogonal Polynomials: The Bessel Polynomials[END_REF]32]) Y m (x) are defined by:

Y m (x) = ∞ k=0 Γ(m + k + 1) k! Γ(m -k + 1) (x/2) k = 2 πx e 1/x K m+ 1 2 (1/x) (3-4)
where K is the modified Bessel function of the second kind [START_REF] Abramowitz | Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables -1974[END_REF]31]. Y m is a polynomial of degree m when m is an integer:

Y 0 = 1 , Y 1 = x + 1 , Y 2 = 3x 2 + 3x + 1 , Y 3 = 15x 3 + 15x 2 + 6x + 1 , etc . . . (3-5)
They satisfy:

x 2 Y ′′ m + (2x + 2)Y ′ m -m(m + 1)Y m = 0 (3-6)
We shall also need:

Q β,j (x) = ∞ k=0 Γ(β + j + k) k! Γ(β -j -k) 2 -k x β-j-k (3-7)
which is a polynomial of degree β -j if β is an integer. In particular Q β,0 is the Carlitz polynomial [START_REF] Carlitz | A Note on the Bessel Polynomials[END_REF]32] and is closely related to Y β-1 :

Q β,0 (x) = x β Y β-1 ( 1 x ) = 2 π e x x β+ 1 2 K β-1 2 (x) (3-8)
satisfying:

x 2 Q ′′ β,0 -2x(β + x)Q ′ β,0 + 2β(x + 1)Q β,0 = 0 (3-9)
The first fews are:

Q 1,0 = x , Q 2,0 = x 2 +x , Q 3,0 = x 3 +3x 2 +3x , Q 4,0 = x 4 +6x 3 +15x 2 +15x , etc . . . (3-10)
For higher j, the Q β,j 's are derivatives of Bessel polynomials:

Q β,j (1/x) = 2 j x j-β d j dx j Y β-1 (x) = 2 j x j-β Y (j) β-1 (x) (3-11)
They satisfy:

-x Q β,j = 1 4 Q β,j+1 + jQ β,j + (j -β)(j + β -1)Q β,j-1 (3-12) Q β,j+1 = 2(β -j -x d dx )Q β,j (3-13)
The first fews are:

Q 2,1 = 2x , Q 3,1 = 6x 2 + 12x , Q 4,1 = 12x 3 + 60x 2 + 90x , (3-14)
Q 3,2 = 24x , Q 4,2 = 120x 2 + 360x , Q 4,3 = 720x , etc . . . (3-15)
3.3 Examples angular integrals with n = 1, 2, 3

• n = 1: The n = 1 case needs no computation, and gives:

I β,1 (x; y) = e xy (3-16)
• n = 2: The n = 2 case requires a little bit of easy computation, and it has been known for some time, we have (this formula is rederived in this article):

I β,2 (X, Y ) = e x 1 y 1 +x 2 y 2 τ β Y β-1 (1/τ ) + e x 1 y 2 +x 2 y 1 (-τ ) β Y β-1 (-1/τ ) = e x 1 y 1 +x 2 y 2 τ 2β Q β,0 (τ ) + e x 1 y 2 +x 2 y 1 (-τ ) 2β Q β,0 (-τ ) (3-17) where τ = - 1 2 (x 1 -x 2 )(y 1 -y 2 ) (3-18)
It can also be written in terms of the modified Bessel function I:

I β,2 (X; Y ) = e 1 2 (x 1 +x 2 )(y 1 +y 2 ) τ 2β-1 I β-1 2 (τ ) , (3-19)
where

I m (τ ) = (τ /2) 2m ∞ k=0 (τ /2) 2k k!Γ(m + k + 1) , I m = I ′′ m + 1 -2m τ I ′ m (3-20)
• n = 3:

we show in this article that (proof in appendix C):

I β,3 ∝ e x 1 y 1 +x 2 y 2 +x 3 y 3 (∆(x)∆(y)) β ∞ k=0 Γ(β -k) 2 6k k! Γ(β + k) i<j Y (k) β-1 ( 1 τ ij ) + perm. (3-21)
where

τ i,j = - (x i -x j )(y i -y j ) 2 (3-22)
and +perm. means that we have to symmetrize over all permutations of the y j 's.

• n > 3: We show in this article that for arbitrary n and β, the angular integral is of the form conjectured by Brezin and Hikami:

I β,n ∝ e i x i y i (∆(x)∆(y)) 2β Îβ,n (τ ij ) + perm. (3-23)
where Îβ,n (τ ij ) is a polynomial in the τ i,j 's, and for which we write a recursion relation.

Transformation of the angular integral

In this section, we transform the Haar measure group integral into a flat Lebesgue measure integral.

Lagrange multipliers

For β = 1/2, 1, 2, an element O ∈ G β,n is an orthonormal basis, i.e. a collection of n orthonormal vectors e 1 , . . . , e n , whose coordinates O i,j = (e i ) j are of the form:

(e i ) j = O i,j = 2β-1 α=0 (e i ) α j ǫ α (4-1)
where the ǫ α 's form a basis of a Clifford algebra (indeed this reproduces the three groups G β,n for β = 1/2, 1, 2):

ǫ 0 = 1 , ǫ † 0 = ǫ 0 , ∀ α > 0 : ǫ 2 α = -1 , ǫ † α = -ǫ α , ǫ α .ǫ α ′ = -ǫ α ′ .ǫ α (4-2)
with structure constants (only for β = 2):

ǫ α ǫ † α ′ = α ′′ η α,α ′ ,α ′′ ǫ α ′′ (4-3)
and where η α,α ′ ,α ′′ has the property useful for our purpose, that for every pair (α, α ′ ), there is exactly only one α ′′ such that η α,α ′ ,α ′′ = 0. In particular η α,α,α ′′ = δ α ′′ ,0 .

The basis must be orthonormal, i.e.

e i .e † j = δ i,j = n k=1 (e i ) k (e j ) † k = n k=1 2β-1 α,α ′ =0 (e i ) α k (e j ) α ′ k ǫ α ǫ † α ′ (4-4)
We introduce Lagrange multipliers to enforce those orthonormality relations

δ(e i .e † i -1) = dS i,i e S i,i (1-k,α ((e i ) α k ) 2 (4-5)
and if i < j:

δ(e i .e † j ) = . . . dS 0 i,j , . . .

dS 2β-1 i,j e -2 α,α ′ ,α ′′ k S α i,j ((e i ) α ′ k )((e j ) α ′′ k )η α ′ ,α ′′ ,α = dS i,j e -2 k S i,j ((e i ) k )((e j ) † k ) (4-6)
where each integral is over the imaginary axis.

Since the scalar product is invariant under group transformations (i.e. change of orthogonal basis), the following measure is invariant and thus must be proportional to the Haar measure:

dO ∝ i,j,α d(e i ) α j i δ(e i .e † i -1) i<j δ(e i .e † j ) (4-7)
i.e.

dO ∝ i,j,α d(e i ) α j dS e i S i,i e -i k S i,i |(e i ) k | 2 e -2 i<j k S α i,j (e i ) k (e j ) † k (4-8)
where

dS = i dS i,i i<j dS i,j = n i=1 dS i,i i<j 2β-1 α=0 dS α i,j (4-9)
is the G β,n invariant measure on the space E β,n :

iS ∈    E 1/2,n = {n × n real symmetric matrices} E 1,n = {n × n hermitian matrices} E 2,n = {n × n quaternion self -dual matrices} (4-10)
where we have completed S by self duality (S = S † ):

S 0 j,i = S 0 i,j
, and

∀ α > 0 S α j,i = -S α i,j (4-11)
Therefore we have (up to a multiplicative constant):

I β,n (X; Y ) ∝ dS de 1 . . . de n e i S i,i e i,k x i y k |(e i ) k | 2 e -i k S i,i |(e i ) k | 2 e -2 i<j k α,α ′ ,α ′′ S α i,j ((e i ) α ′ k )((e j ) † α ′′ k ) η α ′ ,α ′′ ,α (4 -12)
The integral over the (e i ) α k 's is now gaussian and can be performed. The gaussian integrals for each k are independent.

The quadratic form in the exponential is, for each k:

α,α ′ ,α ′′ i,j (e i ) α k (e j ) α ′ k η α,α ′ ,α ′′ (δ i,j x i y k δ α ′′ ,0 -S α ′′ i,j ) (4-13)
If we define the vector

v k = (v 1,k , . . . , v n,k ) where v i,k = α (e i ) α k ǫ † α ,
we have to compute the gaussian integral:

dv k e -v † k (S-y k X)v k (4-14)
For the 3 values of β = 1/2, 1, 2, this integral is worth:

dv k e -v † k (S-y k X)v k = (2π) β det(S -y k X) β (4-15)
where det is the product of singular values (see [START_REF] Mehta | Random Matrices[END_REF]). Thus we get the following theorem:

Theorem 4.1 The angular integral I β,n (X; Y ) is also equal to the following flat Lebesgue measure integral:

I β,n (X; Y ) ∝ dS e Tr S n k=1 det(S -y k X) β = det(X) 1-β dS e Tr SX n k=1 det(S -y k ) β (4-16)
In the last formula we have made the change of variable S → X 1/2 SX 1/2 . Also, the integration domain for S, which was iE β,n before exchanging the integrations over S and e, is now shifted to the right, so that all singular values of (S -y k X) have positive real part. The integration domain for S can be deformed such that the integral remains convergent and the integration path goes to the right of all zeroes of the denominator. If β is half-integer or integer, the denominator is not singular near ∞, and the integration contour can be closed. This will be made more precise below.

Remark 1: For the moment, this formula holds only for β = 1/2, 1, 2. Later we will extend it to other values of β.

Remark 2: Another remark, is that a similar formula can be obtained by exchanging the roles of X and Y .

Duality formula

Notice that the matrix S itself can be diagonalized with a G β,n conjugation:

S = OΛO -1 , Λ = diag(λ 1 , . . . , λ n ) , O ∈ G β,n (4-17)
and the measure dS is up to a constant [START_REF] Mehta | Random Matrices[END_REF]:

dS ∝ dO dΛ ∆(Λ) 2β (4-18)
Therefore, the angular integral reappears in the RHS:

I β,n (X; Y ) ∝ det(X) 1-β dS e Tr SX n k=1 det(S -y k ) β ∝ det(X) 1-β dΛ ∆(Λ) 2β I β,n (X, Λ) n k=1 n j=1 (λ j -y k ) β (4 -19)
Here, if we assume that ∀i, x i ∈ R + , the integration contours for the λ i 's are of the form r + iR where r > max(Re y k ). If β is integer or half integer, the denominator in the integrand is not singular near ∞, and the integration contour can be closed. Thus, if 2β is an integer, the integration contours for the λ i 's can be choosen as circles of radius > max(|y k |).

This equation looks better if we rewrite it in term of the Cauchy determinant D n (X, Y ):

D n (X, Y ) = det 1 x i -y j = ∆(X)∆(Y ) i,j (x i -y j ) (4-20)
and the rescaled function

Ǐβ,n (x 1 , . . . , x n ; y 1 , . . . , y n ) = (∆(X)∆(Y )) β I β,n (x 1 , . . . , x n ; y 1 , . . . , y n ) (4-21)
We then have:

Theorem 4.2 the rescaled function Ǐβ,n satisfies the duality formula:

Ǐβ,n (X; Y ) ∝ det(X) 1-β dΛ Ǐβ,n (X, Λ) D n (Λ, Y ) β (4-22)
i.e. Ǐβ,n (X; Y ) is an eigenfunction of the kernel D β n .

Remark 1: The duality formula above was derived for β = 1/2, 1, 2, but it makes sense for any β.

Remark 2: It is easy to check that this relation is satisfied for the Itzykson-Zuber case β = 1, indeed in that case we have Ǐ1,n (X; Y ) = det(e x i y j ) = ρ (-1) ρ i e y i x ρ(i) , and:

dΛ Ǐ1,n (X, Λ) D n (Λ, Y ) ∝ σ,ρ (-1) σ (-1) ρ n i=1 e λ i x ρ(i) λ i -y σ(i) dλ i = σ,ρ (-1) σ (-1) ρ n i=1 e y σ(i) x ρ(i) = n! det(e x i y j ) ∝ Ǐ1,n (X, Y ) (4-23)

Recursion formula

First, let us notice that we can always assume that x n = 0, otherwise we perform a shift X → X -x n : Thus we define:

I β,n (X, Y ) = G β,n dO e Tr XOY O -1 = G β,
X n-1 = diag(x 1 , . . . , x n-1 ) , X = X n-1 -x n Id n-1 (4-25)
Then, we notice that the orthonormality of the basis e i :

e i .e † j = δ i,j (4-26)
implies that if we already know e 1 , . . . , e n-1 , then e n is completely fixed (up to an irrelevant phase). In other words, it is sufficient to enforce only the orthonormality of e 1 , . . . , e n-1 with Lagrange multipliers, i.e. introduce a matrix S of size n -1. Also, because of our shift X → X -x n , we notice that e n does not appear in the integrand.

Then, we write as in eq.4-8:

dO ∝ n-1 i=1 de i n-1 i=1
δ(e i .e i -1)

n-1 i<j=1 δ(e i .e † j ) ∝ n-1 i=1 n j=1 2β-1 α=0 d(e i ) α j iE β,n-1 dS e i S i,i e -i k S i,i |(e i ) k | 2 e -2 i<j k S i,j (e i ) k (e j ) † k ∝ n-1 i=1 de i iE β,n-1
dS e Tr S e -i,j S i,j e i .e † j (4-27)

which implies, after performing the gaussian integral over the e 1 , . . . , e n-1 :

I β,n (X; Y ) ∝ e xn tr Y iE β,n-1 dS e Tr S n k=1 det(S -y k X) β ∝ e xn tr Y n-1 i=1 (x i -x n ) 2β-1 iE β,n-1 dS e Tr S X n k=1 det(S -y k ) β (4 -28)
Again, S can be diagonalized:

S = OΛO -1 , Λ = diag(λ 1 , . . . , λ n-1 ) , O ∈ G β,n-1 (4-29)
i.e. the rank n angular integral I β,n is expressed in terms of the rank n -1:

I β,n (X; Y ) ∝ e xn tr Y n-1 i=1 (x i -x n ) 2β-1 dΛ I β,n-1 ( X, Λ) ∆(Λ) 2β n k=1 n-1 i=1 (λ i -y k ) β ∝ e xn tr Y n-1 i=1 (x i -x n ) 2β-1 dΛ I β,n-1 (X n-1 , Λ) ∆(Λ) 2β e -xn i λ i n k=1 n-1 i=1 (λ i -y k ) β (4 -30)
Which is our main recursion formula: Theorem 4.3 The angular integrals I β,n (X; Y ) satisfy the recursion:

I β,n (X; Y ) ∝ e xn n i=1 y i n-1 i=1 (x i -xn) 2β-1 dλ 1 , . . . dλ n-1 I β,n-1 (X n-1 ,Λ) ∆(Λ) 2β e -xn i λ i n k=1 n-1 i=1 (λ i -y k ) β (4-31)
Here again, the integration contours for the λ i 's are such that the integral is convergent, and such that they surround all the y k 's. For instance, if 2β is an integer, and if ∀i x i ∈ R + , the integration contour for the λ i 's can be choosen as circles of radius > max(|y k |).

Remark 4.1 Now this recursion formula can be used to define I β,n for arbitrary β, so that it coincides with the angular integral for β = 1/2, 1, 2.

We have also the iterated form:

I β,n (X; Y ) ∝ e xn n i=1 y i ∆(X) 2β-1 n-1 i=1 i j=1 dλ i,j n-1 i=1 1≤j<j ′ ≤i (λ i,j ′ -λ i,j ) 2β n-1 i=1 e (x i -x i+1 ) j λ i,j n-1 i=1 i+1 j=1 i j ′ =1 (λ i,j ′ -λ i+1,j ) β (4 -32)
where we have defined λ n,j = y j , and where the integration contours are circles such that:

|λ i,j | = ρ i , ρ 1 > ρ 2 > . . . > ρ n-1 > max|y j | (4-33)
Remark 4.2 A similar recursion relation was also found Kohler and Guhr [15,16,[START_REF] Kohler | Supersymmetric extensions of Calogero-Moser-Sutherlandlike models: construction and some solutions[END_REF],

but the authors found real integrals instead of contour integrals. The advantage of our formulation, is that we can easily move integration contours and find new relations, as we will see below.

Moments of angular integrals and Calogero

In this section, we compute moments of the angular integral, and we show that our formula indeed satisfies Calogero equation.

Generalized Morozov's formula

Define the quadratic moments (see [28] for β = 1):

M i,j = G β,n dO ||O i,j || 2 e Tr XOY O -1 (5-1)
The same calculation as above yields:

M i,j = β dS e Tr S n k=1 det(S -y k X) β ((S -y j X) -1 ) i,i = β x i det(X) 1-β dS e Tr SX n k=1 det(S -y k ) β ((S -y j ) -1 ) i,i (5 -2)
As a consistency check, and as a warmup exercise, let us show that this formula satisfies:

I β,n = j M i,j (5-3) which comes from ∀O ∈ G β,n , j ||O i,j || 2 = 1.
We have:

j M i,j = j β x i det(X) 1-β dS e Tr SX n k=1 det(S -y k ) β ((S -y j ) -1 ) i,i = -1 x i det(X) 1-β dS e Tr SX ∂ ∂S i,i 1 n k=1 det(S -y k ) β = 1 x i det(X) 1-β dS 1 n k=1 det(S -y k ) β ∂ ∂S i,i e Tr SX = det(X) 1-β dS 1 n k=1 det(S -y k ) β e Tr SX = I β,n (5-4)
Notice that this equality holds independently of the integration domain of S, provided that one can integrate by parts without picking boundary terms.

Remark: Of course a similar equation can be found by exchanging the roles of X and Y , and one gets symmetrically:

i M i,j = I β,n
(5-5)

Other moments

Since, after introducing the Lagrange multipliers, the integral becomes gaussian in the O i,j 's, any polynomial moment can be computed using Wick's theorem. It is sufficient to compute the propagator:

< O i,k O † j,l >= β δ k,l (S -y k X) -1 i,j (5-6)
Then, the expectation value of any polynomial moment is obtained as the sum over all pairings of the product of propagators.

For instance:

dO e Tr XOY O -1 O i 1 ,j 1 O i 2 ,j 2 O † i 3 ,j 3 O † i 4 ,j 4 = dS e Tr S n k=1 det(S -y k X) β δ j 1 ,j 3 δ j 2 ,j 4 (S -y j 1 X) -1 i 1 ,i 3 (S -y j 2 X) -1 i 2 ,i 4 +δ j 1 ,j 4 δ j 2 ,j 3 (S -y j 1 X) -1 i 1 ,i 4 (S -y j 2 X) -1 i 2 ,i 3 (5 -7)
In principle, one could compute with this method the generalization of all Shatashvili's moments [30].

Linear equations

In this section we prove that the M i,j 's satisfy the following linear functional relations, which are very similar to Dunkl equations []:

∀i, j , ∂M i,j ∂y j + β l =j M i,l -M i,j y l -y j = M i,j x i (5-8)
We are going to give 2 different proofs of eq. [START_REF] Carlitz | A Note on the Bessel Polynomials[END_REF][START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF][START_REF] Desrosiers | Duality in random matrix ensembles for all Beta[END_REF][START_REF] Duistermaat | [END_REF]. The first one below is based on integration by parts. It can be done for the 3 groups β = 1/2, 1, 2, however it is rather tedious for β = 1/2 and β = 2, and we present the proof only for β = 1. Another proof valid for all 3 values of β is presented in section 5.6 below.

Let us check that eq. (5-2) satisfies eq. (5-8) (for β = 1). We first rewrite:

1 y l -y j 1 S -y l - 1 S -y j i,i = ((S -y l ) -1 (S -y j ) -1 ) i,i = m ((S -y l ) -1 ) i,m ((S -y j ) -1 ) m,i (5-9)
For β = 1, we may consider all variables S i,m to be independent variables, and we integrate by parts:

l =j M i,l -M i,j y l -y j = l =j m β x i det(X) 1-β dS e Tr SX n k=1 det(S -y k ) β ((S -y l ) -1 ) i,m ((S -y j ) -1 ) m,i = - m 1 x i det(X) 1-β dS e Tr SX det(S -y j ) β ((S -y j ) -1 ) m,i ∂ ∂S i,m 1 l =j det(S -y l ) β = m 1 x i det(X) 1-β dS 1 l =j det(S -y l ) β ∂ ∂S i,m e Tr SX det(S -y j ) β ((S -y j ) -1 ) m,i = m 1 x i det(X) 1-β dS e Tr SX k det(S -y k ) β ((S -y j ) -1 ) m,i x i δ i,m -β m 1 x i det(X) 1-β dS e Tr SX k det(S -y k ) β ((S -y j ) -1 ) i,m ((S -y j ) -1 ) m,i - m 1 x i det(X) 1-β dS e Tr SX k det(S -y k ) β ((S -y j ) -1 ) m,m ((S -y j ) -1 ) i,i = det(X) 1-β dS e Tr SX k det(S -y k ) β ((S -y j ) -1 ) i,i - 1 x i det(X) 1-β dS e Tr SX k det(S -y k ) β ((S -y j ) -2 ) i,i - β x i det(X) 1-β dS e Tr SX k det(S -y k ) β Tr (S -y j ) -1 ((S -y j ) -1 ) i,i = 1 β x i M i,j - ∂M i,j ∂y j (5-10)
QED. The same computation can be repeated for β = 1/2 and β = 2, with additional steps because the variables S i,m are no longer independent, and also because for β = 2, det(S -y j ) is defined as the product of singular values. Another proof is given in section 5.6.

Remark: Of course a similar equation can be found by exchanging the roles of X and Y , and one gets the symmetric linear equation:

∀i, j , ∂M i,j ∂x i + β l =i M l,j -M i,j x l -x i = M i,j y j (5-11)
Remark: again, this proves that eq. (5-2) is solution of the differential equation eq. [START_REF] Carlitz | A Note on the Bessel Polynomials[END_REF][START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF][START_REF] Desrosiers | Duality in random matrix ensembles for all Beta[END_REF][START_REF] Duistermaat | [END_REF], for any choice of integration domain provided that we can integrate by parts. In fact, by taking linear combinations of all possible integration contours, we get the general solution of the linear equation eq. [START_REF] Carlitz | A Note on the Bessel Polynomials[END_REF][START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF][START_REF] Desrosiers | Duality in random matrix ensembles for all Beta[END_REF][START_REF] Duistermaat | [END_REF]. However, a general solution of eq. (5-8) is not necessarily symmetric in X and Y , and does not necessarily obey eq. (5-11).

Calogero equation

Here, we prove that I β,n satisfies the Calogero equation.

Start from the linear equation:

∂M i,j ∂x i + β k =i M i,j -M k,j x i -x k = M i,j y j (5-12)
then sum over j, using eq. (5-3):

∂I ∂x i = j M i,j y j (5-13)
Then apply ∂ ∂x i :

∂ 2 I ∂x 2 i = j y j ∂M i,j ∂x i = j y j M i,j y j -β k =i M i,j -M k,j x i -x k = j y 2 j M i,j -β j y j k =i M i,j -M k,j x i -x k = j y 2 j M i,j -β k =i ∂I ∂x i -∂I ∂x k x i -x k (5 -14)
If we take the sum over i, using eq. (5-5), we get:

i ∂ 2 I ∂x 2 i + β i k =i ∂I ∂x i -∂I ∂x k x i -x k = j y 2 j I (5-15)
i.e. we recover the Calogero equation:

H Calogero .I β,n = ( j y 2 j ) I β,n
(5-16)

Matrix form of the linear equations

The linear equations, are n 2 linear equations of order 1, for n 2 unknown functions M i,j .

They can be summarized into a matricial equation:

M Y = K M (5-17)
where K is a matricial operator

K ii = ∂ ∂x i + β k =i 1 (x i -x k ) , K ik = - β (x i -x k ) i = k (5-18)
and more generally this implies:

M Y p = K p M (5-19)
and therefore, for any polynomial P :

M.P (Y ) = P (K).M (5-20)
In particular if we choose the characteristic polynomial of Y : And if P (K) = l =j (y l -K), we get:

0 = n i=1 (y i -K) . M (
M i,j = m l =j y l -K y l -y j i,m . I β,n (5-26)
This last relation allows to reconstruct M i,j if we know I β,n .

Finally, before leaving this section, we just mention that those operators K i,j are also related to the Laplacian over the set of matrices E β,n , as was noted recently by Zuber [START_REF] Zuber | On the large N limit of matrix integrals over the orthogonal group[END_REF].

Linear equation from loop equations

There is another way of deriving those Dunkl-like linear equations for the angular integrals, using loop equations of an associated 2-matrix model.

Consider the following 2-matrix integral, where M 1 and M 2 are both in the E β,n ensemble:

Z = dM 1 dM 2 e -Tr (V 1 (M 1 )+V 2 (M 2 )-M 1 M 2 ) (5-27) After diagonalization of M 1 = O 1 XO -1 1 and M 2 = O 2 Y O -1
2 , we have:

Z = dXdY dO 1 dO 2 ∆(X) 2β ∆(Y ) 2β e -Tr (V 1 (X)+V 2 (Y )) e Tr XO -1 1 O 2 Y O -1 2 O 1
(5-28)

We redefine O 2 = O 1 .O, and the integral over O 1 gives 1, and the integral over O gives the angular integral:

Z = dXdY ∆(X) 2β ∆(Y ) 2β e -Tr (V 1 (X)+V 2 (Y )) I β,n (X, Y ) (5-29)
We can do a similar change of variable for moments:

< Tr 1 x -M 1 1 y -M 2 > = 1 Z dM 1 dM 2 e -Tr (V 1 (M 1 )+V 2 (M 2 )-M 1 M 2 ) Tr 1 x -M 1 1 y -M 2 = 1 Z dXdY dO 1 dO 2 ∆(X) 2β ∆(Y ) 2β e -Tr (V 1 (X)+V 2 (Y ))
e Tr XO -1

1 O 2 Y O -1 2 O 1 Tr 1 x -X O -1 1 O 2 1 y -Y O -1 2 O 1 = 1 Z dXdY ∆(X) 2β ∆(Y ) 2β e -Tr (V 1 (X)+V 2 (Y )) i,j M i,j (X, Y ) 1 x -x i 1 y -y j (5 -30)
where M i,j (X, Y ) is the Morozov moment defined in eq.5-1.

Loop equations amount to say that an integral is invariant under a change of variables. Thus, we change

M 1 → M 1 + ǫ 1 x-M 1 1 y-M 2 + O(ǫ 2
) in Z, and to order 1 in ǫ we get (the loop equations for β = 1/2, 1, 2 ensembles can be found in several references [], the Jacobian is easily computed in eigenvalue representation, see appendix B, eq. (2-8)):

0 = Tr 1 x -M 1 M 2 y -M 2 -Tr V ′ 1 (M 1 ) x -M 1 1 y -M 2 +β Tr 1 x -M 1 Tr 1 x -M 1 1 y -M 2 +(β -1) ∂ ∂x Tr 1 x -M 1 1 y -M 2 (5 -31) i.e., going to eigenvalues M 1 = O 1 XO -1 1 and M 2 = O 2 Y O -1 2 : 0 = i,j (y j -V ′ 1 (x i )) M i,j (X, Y ) (x -x i )(y -y j ) +β i =l j M i,j (X, Y ) (x -x l )(x -x i )(y -y j ) + i,j M i,j (X, Y ) (x -x i ) 2 (y -y j ) (5 -32)
The last term can be integrated by parts:

i,j M i,j (X, Y ) (x -x i ) 2 (y -y j ) = i,j dX dY ∆(X) 2β ∆(Y ) 2β e -Tr (V 1 (X)+V 2 (Y )) M i,j (X, Y ) ∂ ∂x i 1 (x -x i ) (y -y j ) = - i,j dX dY 1 (x -x i ) (y -y j ) ∂ ∂x i M i,j (X, Y ) ∆(X) 2β ∆(Y ) 2β e -Tr (V 1 (X)+V 2 (Y )) = i,j dX dY M i,j (X, Y ) (x -x i ) (y -y j ) ∆(X) 2β ∆(Y ) 2β e -Tr (V 1 (X)+V 2 (Y )) V ′ 1 (x i ) - l =i 2β x i -x l - i,j dX dY 1 (x -x i ) (y -y j ) ∆(X) 2β ∆(Y ) 2β e -Tr (V 1 (X)+V 2 (Y )) ∂M i,j (X, Y ) ∂x i = i,j V ′ 1 (x i )M i,j (X, Y ) (x -x i )(y -y j ) -2β l =i j M i,j (X, Y ) (x i -x l )(x -x i )(y -y j ) - i,j 1 (x -x i )(y -y j ) ∂M i,j (X, Y ) ∂x i
(5 -33)

Therefore we have:

i,j 1 (x -x i )(y -y j ) ∂M i,j (X, Y ) ∂x i = i,j y j M i,j (X, Y ) (x -x i )(y -y j ) -2β l =i j M i,j (X, Y ) (x i -x l )(x -x i )(y -y j ) +β i =l j M i,j (X, Y ) (x -x l )(x -x i )(y -y j ) = i,j y j M i,j (X, Y ) (x -x i )(y -y j ) -2β l =i j M i,j (X, Y ) (x i -x l )(x -x i )(y -y j ) +β i =l j M i,j (X, Y ) (x -x l )(x l -x i )(y -y j ) +β i =l j M i,j (X, Y ) (x -x i )(x i -x l )(y -y j ) = i,j y j M i,j (X, Y ) (x -x i )(y -y j ) -β l =i j 1 (x -x i )(y -y j ) M i,j (X, Y ) -M l,j (X, Y ) (x i -x l )
(5 -34)

Since this equation must hold for any V 1 and V 2 , x, y, i.e. for any measure on X and Y , it must hold term by term i.e. we recover the linear equation:

∂M i,j (X, Y ) ∂x i = y j M i,j (X, Y ) -β l =i M i,j (X, Y ) -M l,j (X, Y ) (x i -x l ) (5-35)
Of course, the loop equation coming from the change of variable

M 2 → M 2 + ǫ 1 x-M 1 1 y-M 2 + O(ǫ 2 )
gives the symmetric linear equation:

∂M i,j (X, Y ) ∂y j = x i M i,j (X, Y ) -β l =j M i,j (X, Y ) -M i,l (X, Y ) (y j -y l )
(5-36) QED.

Principal terms and the τ ij variables

As we mentioned in the introduction, it was noticed in particular by Brezin and Hikami [START_REF] Brezin | An extension of the Harish-ChandraItzykson-Zuber integral[END_REF], that the angular integral can be written as combinations of exponential terms, and polynomials (for β integer, series otherwise), of some reduced variables τ i,j = -1 2 (x i -x j )(y i -y j ). Here, we show how our recursion gives such a form. We thus define: Definition 6.1 We define the principal term Îβ,n (X; Y ) from the recursion:

Îβ,1 = 1 and Îβ,n (X n ; Y n ) = ∆(Y n ) 2β n-1 i=1 (x i -x n ) Res λ i →y i dλ 1 (λ 1 -y 1 ) β . . . dλ n-1 (λ n-1 -y n-1 ) β Îβ,n-1 (X n-1 , Λ) e i (x i -xn)(λ i -y i ) n k=1 n-1 i=1, =k (y k -λ i ) β (6 -1)
It is such that (the sum over permutations comes from the sum of residues at all poles in recursion eq. (4-31) of theorem. 4.3):

I β,n (X, Y ) = σ e n i=1 x i y σ(i) ∆(X) 2β ∆(Y σ ) 2β Îβ,n (X, Y σ ) (6-2)
In [START_REF] Brezin | An extension of the Harish-ChandraItzykson-Zuber integral[END_REF], Brezin and Hikami observed and conjectured that when β is an integer, Îβ,n (X, Y ) is a polynomial in the variables τ i,j = -1 2 (x i -x j )(y i -y j ).

For instance, if β = 1, we have for arbitrary n:

Î1,n = i<j τ i,j (6-3)
And, if n = 2, we have for arbitrary β:

Îβ,2 = (x 1 -x 2 ) (y 1 -y 2 ) 2β Res λ→0 dλ λ β e (x 1 -x 2 )λ (y 2 -y 1 -λ) β = (x 1 -x 2 ) (y 1 -y 2 ) 2β ∂ β-1 ∂λ β-1 e (x 1 -x 2 )λ (y 2 -y 1 -λ) β λ=0 = (x 1 -x 2 ) (y 1 -y 2 ) 2β β-1 k=0 (β -1)! k!(β -1 -k)! (x 1 -x 2 ) β-1-k (y 2 -y 1 ) -β-k (β -1 + k)! (β -1)! = β-1 k=0 (β -1 + k)! k!(β -1 -k)! ((x 1 -x 2 )(y 2 -y 1 )) β-k = 2 β Q β,0 (τ 1,2 ) (6-4)
i.e. we recover the well known result that Îβ,2 is the Bessel polynomial of degree β.

We are going to prove the conjecture of Brezin Hikami for all n and for all β ∈ N, but first, let us prove some preliminary properties: Lemma 6.1 Îβ,n is a polynomial in all variables x i and y j , and it is symmetric under the exchange X ↔ Y , and under the permutation of pairs (x i , y i ) ↔ (x j , y j ), and under translations X → X + cte.Id or Y → Y + cte.Id.

proof:

If β is an integer, the recursion relation eq. (6-1) leads to (we write x i,j = x i -x j , y i,j = y i -y j ):

Îβ,n+1 = n i=1 x i,n+1 y β n+1,i ∂ ∂λ i β-1 e x i,n+1 λ i 1≤j≤n+1,j =i y β j,i (y j,i -λ i ) β Îβ,n (X n , Y n + Λ) Λ=0 (6 -5)
which shows by recursion, that Îβ,n+1 is a rational function of all x i 's and y j 's. We know, from its very definition, that the angular integral

I β,n (X, Y ) = σ e n i=1 x i y σ(i) ∆(X) 2β ∆(Y σ ) 2β Îβ,n (X, Y σ ) (6-6)
is symmetric in all x i 's and y j 's, and in the exchange X ↔ Y . Since the exponentials are linearly independent on the ring of rational functions, each term must be symmetric in permutations of pairs (x i , y i )'s, i.e. Îβ,n (x 1 , . . . , x n ; y 1 , . . . , y n ) is a symmetric function of the pairs (x i , y i )'s, and also symmetric under X ↔ Y . Moreover, Îβ,n+1 is clearly a polynomial in the variables x n+1 and y n+1 , and because of the symmetry, it must also be a polynomial in all variables. Translation invariance is also clear from the recursion formula. Îβ,n is a symmetric polynomial of degree β in the τ i,j 's.

proof: Using lemma 6.1, it is easy to see that Îβ,n fulfills the hypothesis of lemma A.3 in the appendix A, and this proves the theorem.

I.e. we have proved the conjecture of Brezin and Hikami [START_REF] Brezin | An extension of the Harish-ChandraItzykson-Zuber integral[END_REF]. In fact, we notice that the property of being a polynomial in the τ 's, is not specific to angular integrals, but comes only from the global symmetries.

Recursion without residues for β integer

For β ∈ N, the residues in recursion eq. (6-1) can be performed, and they compute derivatives of the integrand. Thus eq. (6-1) can be rewritten:

Îβ,n (X n ; Y n ) = ∆(Y n ) 2β n-1 i=1 (x i -x n ) Res λ i →a i dλ 1 (λ 1 -a 1 ) β . . . dλ n-1 (λ n-1 -a n-1 ) β Îβ,n-1 (X n-1 , Λ) e i (x i -xn)(λ i -y i ) n k=1 n-1 i=1, =k (y k -λ i ) β a i =y i = ∆(Y n ) 2β (β -1)! n-1 n-1 i=1 (x i -x n ) i ∂ ∂a i β-1 Res λ i →a i dλ 1 (λ 1 -a 1 ) . . . dλ n-1 (λ n-1 -a n-1 ) Îβ,n-1 (X n-1 , Λ) e i (x i -xn)(λ i -y i ) n k=1 n-1 i=1, =k (y k -λ i ) β a i =y i (6 -7)
i.e. we can perform the residues:

Îβ,n (X n ; Y n ) = ∆(Y n ) 2β (β -1)! n-1 n-1 i=1 x i,n ∂ ∂a i β-1 Îβ,n-1 (X n-1 , a) e i x i,n (a i -y i ) n k=1 n-1 i=1, =k (y k -a i ) β a i =y i (6-8) More explicitly Îβ,n (X n ; Y n ) = n-1 i=1 β-1 γ i ,β i,k =0 (x i,n y n,i ) β i,n Γ(β -γ i -k β i,k ) β i,n ! n-1 k=1,k =i Γ(β + β i,k ) β i,k ! Γ(β) 1 (x i,n y k,i ) β i,k 1 γ i ! 1 x i,n ∂ ∂a i γ i Îβ,n-1 (X n-1 , a) a i =y i (6 -9)

n = 3

For n = 3, and arbitrary β, we prove that:

Theorem 6.2 Îβ,3 = ∞ k=0 Γ(β -k) 2 3k k! Γ(β + k) i<j Q β,k (τ ij ) (6-10)
In fact, for β integer, the sum over k is finite and reduces to k ≤ β -1.

The proof, rather technical, is given in appendix C. We used the Calogero equation.

Conjecture for higher n

Applying the recursion relations of this article, we also computed the n = 4 case for small values of β:

Î2,4 = i<j Q 2,0 (τ ij ) + 1 16 Q 2,0 (τ 1,2 )Q 2,0 (τ 1,3 )Q 2,0 (τ 1,4 ) Q 2,1 (τ 2,3 )Q 2,1 (τ 2,4 )Q 2,1 (τ 3,4 ) + sym + 1 128 Q 2,0 (τ 1,2 ) Q 2,1 (τ 1,3 )Q 2,1 (τ 1,4 )Q 2,1 (τ 2,3 )Q 2,1 (τ 2,4 )Q 2,1 (τ 3,4 ) + sym (6 -11) and: Î3,4 = 64 i<j Q 3,0 (τ ij ) + 4 3 Q 3,0 (τ 1,2 )Q 3,0 (τ 1,3 )Q 3,0 (τ 1,4 ) Q 3,1 (τ 2,3 )Q 3,1 (τ 2,4 )Q 3,1 (τ 3,4 ) + sym + 1 48 Q 3,0 (τ 1,2 )Q 3,0 (τ 1,3 )Q 3,0 (τ 1,4 ) Q 3,2 (τ 2,3 )Q 3,2 (τ 2,4 )Q 3,2 (τ 3,4 ) + sym + . . . (6-12)
Those expressions lead us to conjecture a general form in terms of Bessel polynomials: Conjecture 6.1 We conjecture that for all n and β, Îβ,n is of the form:

Îβ,n = {l} A {l} i<j Q β,l (i,j) (τ i,j ) (6-13)
Unfortunately, we have not been able so far to determine the general form of the coefficients A {l} for n > 3 (except n = 4 and β = 2).

Symplectic case β = 2

For β = 2, the recursion eq. (6-8), reduces to:

Î2,n = ∆(Y n ) 4 n-1 i=1 x i,n i ∂ ∂a i Î2,n-1 (X n-1 , a) e i (x i -xn)(a i -y i ) n k=1 n-1 i=1, =k (y k -a i ) 2 a i =y i = n-1 i=1 x i,n y 2 i,n x i -x n - n k=1, =i 2 y i -y k + ∂ ∂a i Î2,n-1 (X n-1 , a) a i =y i (6 -14)
From a recursion hypothesis, we assume that Î2,n-1 (X n-1 , a) is a polynomial in the τ 's of the form:

Î2,n-1 (X n-1 , Y n-1 ) = {l} A {l} i<j Q 2,l i,j (τ i,j ) (6-15)
where for every pair (i, j) we have l i,j ∈ {0, 1}. We recall that:

|0 > τ = Q 2,0 (τ ) = τ 2 + τ , |1 > τ = Q 1,0 (τ ) = 2τ (6-16)
Thus we may write:

∂ ∂a i = - 1 2 k =i x i,k ∂ ∂t i,k , t i,k = - 1 2 (x i -x k )(a i -a k ) (6-17)
and we define the operators C i,k acting on functions of the variable τ i,k such that:

C i,k = 1 τ i,k - 1 2 ∂ ∂t i,k (6-18)
and all derivatives must be eventually computed at t i,k = τ i,k . Since our operators act on expressions of the form eq. (6-15), we need to compute:

C i,j |0 >= C i,j .Q 2,0 (τ i,j ) = C i,j .(τ 2 i,j + τ i,j ) = 1 2 (6-19) C i,j |1 >= C i,j .Q 2,1 (τ i,j ) = C i,j . 2τ i,j = 1 (6-20)
And we may also have terms of the form C i,j .C j,i , for which we have:

C i,j C j,i |0 >= C i,j C j,i .(τ 2 i,j + τ i,j ) = - 1 2 , C i,j C j,i |1 >= C i,j C j,i .τ i,j = 0 (6-21)
Finally we have:

Î2,n = n-1 i=1 τ 2 i,n n-1 i=1 1 + 1 τ i,n + n-1 k=1, =i x i,k x i,n C i,k Î2,n-1 t i,j =τ i,j (6-22)
It is more convenient to rewrite this in terms of a Hilbert space with basis |0 >= Q 2,0 and |1 >= Q 2,1 , and thus:

Î2,n = n-1 i=1 τ 2 i,n + τ i,n + 1 2 τ i,n n-1 k=1, =i y n,i y k,i A i,k Î2,n-1 (6 -23)
where

A i,k = 2τ i,k C i,k (6-24)
We have:

A|0 >= 1 2 |1 > , A|1 >= |1 > (6-25)
and:

A i,k A k,i = 2(A i,k -1) (6-26)
Unfortunately we have not been able to go further with this formulation.

Triangle conjecture

We have seen that there is an operator formalism for computing angular integrals, with operators A i,j associated to "edges" (i, j). However, it can be seen for n = 3, 4, that operator edges appear only in certain combinations, which involve triangles (i, j, k).

We thus introduce the triangles operators:

T i,j,k = π i,j π j,k π k,i (6-27)
where π i,j = π j,i is the projector on state 1 2 |1 > i,j = τ i,j :

π i,j .(τ 2 i,j + τ i,j ) = τ i,j , π i,j .τ i,j = τ i,j , π i,j = π j,i , π 2 i,j = π i,j (6-28
) With this notation we have:

Î2,3 = (1 + 1 2 T 1,2,3 ). 1≤i<j≤3 |0 > i,j (6-29)
where we recall that |0 > i,j = τ 2 i,j + τ i,j . And

Î2,4 = 1 + 1 2 (T 1,2,3 + T 1,2,4 + T 1,3,4 + T 2,3,4 ) + 1 4 (T 1,2,3 T 1,2,4 + T 1,2,3 T 1,3,4 + T 1,2,4 T 1,3,4 +T 1,2,3 T 2,3,4 + T 1,2,4 T 2,3,4 + T 1,3,4 T 2,3,4 ) . 1≤i<j≤4 |0 > i,j (6-30)
We are naturally led to conjecture that:

Î2,n = triangulations T C T T ∈T . i<j |0 > i,j (6-31)
We have:

C ∅ = 1 , C (i,j,k) = 1 2 , C (i,j,k),(i,j,l) = 1 4 , . . . (6-32)
We have not been able so far to prove this conjecture. It is to be noted from the low values of n, that triangles seem to play a role for all β ∈ N.

Additional results: determinantal recursion

Just for completeness, we give another form of the recursion eq. [START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF][START_REF] Desrosiers | Duality in random matrix ensembles for all Beta[END_REF][START_REF] Duistermaat | [END_REF][START_REF] Dyson | Statistical Theory of the Energy Levels of Complex Systems. I[END_REF](10)[START_REF] Eynard | A short note about Morozovs formula[END_REF][START_REF] Eynard | Asymptotics of skew orthogonal polynomials[END_REF](13)(14), in terms of determinants:

Î2,n+1 (X n+1 ; Y n+1 ) = det X -x n+1 -2 Y -y n+1 + B + ∂ Y det(X -x n+1 ) Î2,n (X n ; Y n ) (6-33) where Î2,n (X, Y ) = 1 i<j τ 2 i,j Î2,n (X, Y ) (6-34)
and where B is the antisymmetric matrix

B ij = √ 2 y i -y j (6-35)
and ∂ Y = diag(∂ y 1 , . . . , ∂ yn ). This is proved by observing that the expansion of the determinant eq. (6-33) can be interpreted like a Wick's expansion equivalent to eq. [START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF][START_REF] Desrosiers | Duality in random matrix ensembles for all Beta[END_REF][START_REF] Duistermaat | [END_REF][START_REF] Dyson | Statistical Theory of the Energy Levels of Complex Systems. I[END_REF](10)[START_REF] Eynard | A short note about Morozovs formula[END_REF][START_REF] Eynard | Asymptotics of skew orthogonal polynomials[END_REF](13)(14).

Conclusion

In this article, we have found many new relations and new representations of angular integrals.

First, we have been able to rewrite angular integrals with a complicated Haar measure, in terms of usual Lebesgue measure contour integrals. Then, we have deduced duality and recursion formulae.

This allowed us to prove Brezin-Hikami's conjecture, and to find some explicit form for n = 3, and conjecture some explicit form in terms of Bessel polynomials for the general case.

For β = 2, we have simplified our recursion (computed the residues). The same method seems to be applicable for higher β, but we have not done it in this article.

We have obtained many new forms of angular integrals, but unfortunately, this does not seem to be the end of the story. Our expressions are still not explicit enough to be useful for computing matrix integrals. The form of our expressions, strongly suggest that the kernel determinantal formulae [START_REF] Mehta | Random Matrices[END_REF] in the β = 1 case, could be replaced by hyperdeterminantal formulae for higher β, but this is still to be understood. The best thing, would be to get expressions with enough structure to generalize the method over integration of matrix variables of Mehta [START_REF] Mehta | A method of integration over matrix variables[END_REF].

It clearly holds for n = 0 and n = 1. Indeed the n = 1 case reads:

x 1 y 2 + x 2 y 1 = x 1 y 1 + x 2 y 2 -(x 1 -x 2 )(y 1 -y 2 ) = 2(τ 1,2 -τ 1,3 -τ 2,3 ) (1-3)
Assume that the lemma holds up to n -1, and let us prove it for n. In the following A ≡ B means that A -B is a polynomial in the τ 's. Therefore P n ≡ -P n , i.e. P n is a polynomial in the τ 's.

P n = x 1 x 2 . . . x
Lemma A.2 Let P α,β (X, Y ) = n i=1 x α i i y β i i + n i=1 x β i i y α i i , i α i = i β i = d (1-10)
We prove that P α,β is a polynomial of degree d in the τ 's, with integer coefficients: -11) proof:

P n ∈ Z[τ ] ( 1
We proceed by recursion on the total degree d = i α i = i β i . The lemma clearly holds for d = 0 and d = 1. Assume d ≥ 2 and that the lemma holds up to d -1. We will prove it for d.

• if there exists i such that α i β i > 0, then P α,β is the factor of τ i,n+1 times P α ′ ,β ′ of smaller degree, and from the recursion hypothesis it holds.

• assume that ∀i, α i β i = 0. Since d ≥ 2, there must exist some i such that α i ≥ 1 and some j such that β j ≥ 1. Let us choose k and l such that:

α k = max i {α i } ≥ 1 , β l = max i {β i } ≥ 1 (1-12)
We have k = l.

• 1-matrix model in eigenvalue representation, for arbitrary β: Consider the integral:

Z = dλ 1 . . . dλ n e -i V (λ i ) i<j (λ j -λ i ) 2β (2-1)
If we make an infinitesimal local change of variable λ i → λ i + ǫλ k i + O(ǫ 2 ), we find:

Z = (1 + O(ǫ 2 )) dλ 1 . . . dλ n i (1 + ǫkλ k-1 i ) e -i V (λ i ) i (1 -ǫλ k i V ′ (λ i )) i<j (λ j -λ i ) 2β i<j (1 + 2βǫ λ k i -λ k j λ i -λ j ) (2 -2)
i.e., by considering the term linear in ǫ:

0 = i kλ k-1 i + β k-1 l=0 i =j λ l i λ k-1-l j - i λ k i V ′ (λ i ) (2-3)
It can be written collectively by summing over k with 1/x k+1 , this is equivalent to consider a local change of variable λ i → λ i + ǫ x-λ i +O(ǫ 2 ), and we write ω

(x) = i 1 x-λ i : 0 = -ω ′ (x) + β(ω(x) 2 + ω ′ (x)) - i V ′ (λ i ) x -λ i (2-4) i.e. 0 = -Tr V ′ (M) x -M + β Tr 1 x -M Tr 1 x -M + (β -1) ∂ ∂x Tr 1 x -M (2-5)
• 2-matrix model for β = 1/2, 1, 2: Similarly if we consider a 2-matrix model:

Z = dM 1 dM 2 e -Tr (V 1 (M 1 )+V 2 (M 2 )-M 1 M 2 ) (2-6)
again we make a local change of variable

M 1 → M 1 + ǫ 1 x-M 1 A + O(ǫ 2
). The Jacobian of this change of variable is computed as a split rule (cf []), it can be computed for each of the 3 ensembles β = 1/2, 1, 2 and is worth:

dM 1 → dM 1 1 + ǫβ Tr 1 x -M 1 Tr A 1 x -M 1 + ǫ(β -1) ∂ ∂x Tr A 1 x -M 1 + O(ǫ 2 )
(2-7) Thus we find:

0 = -Tr (V ′ 1 (M 1 ) -M 2 ) 1 x -M 1 + β Tr 1 x -M 1 Tr A 1 x -M 1 +(β -1) ∂ ∂x Tr A 1 x -M 1 (2-8)
• Without a definition of a 2-matrix integral for arbitrary β, it is not possible to find the loop equation for any β. However, we see that equation eq. (2-8) is valid for the 2-matrix model for β = 1/2, 1, 2, and is valid for the 1-matrix eigenvalue model for any β. Therefore it is natural to take it as a definition of the 2-matrix model for arbitrary β.

C Appendix:

Proof n = 3

C.1 BESSEL ZOOLOGY

We define the functions

Q β (x) = ∞ l=0 (-) l Γ (β + l) Γ (β -l) 2 -l l! x β-l (3-1)
We have

2x 2 Q ′ β (x) -2β x Q β (x) = - ∞ l=1 (-) l Γ (β + l) Γ (β -l) 2 -l+1 (l -1) x β-l+1 x 2 Q " β (x) -2β x Q ′ β (x) + 2β Q β (x) = - ∞ l=0 (-) l Γ (β + l + 1) Γ (β -l -1) 2 -l l! x β-l (3-2)
Changing l into l + 1 in 3-2 we obtain the differential equation

x 2 Q " β (x) -2x (β -x) Q ′ β (x) + 2β (1 -x) Q β (x) = 0 (3-3)
We define the functions

Q β,k (x) = (-2) k x β-k x 2 d dx k Q β (x) x β (3-4) Q β,0 (x) = Q β (x) (3-5) that is Q β,k (x) = ∞ l=0 (-) l+k Γ (β + l + k) Γ (β -l -k) 2 -l l! x β-l-k (3-6)
From 3-4 we obtain the recurrence

Q β,k (x) = 2 (β -k + 1) Q β,k-1 (x) -2x Q ′ β,k-1 (x) (3-7) For instance Q β,1 (x) = 2β Q β,0 (x) -2x Q ′ β,0 (x) (3-8) and Q β,2 (x) = 2 (β -1) Q β,1 (x) -2x Q ′ β,1 (x) (3-9) Q β,2 (x) = 4x 2 Q " β,0 (x) -8x (β -1) Q ′ β,0 (x) + 4β (β -1) Q β,0 (x) (3-10)
Thus, we have, from 3-8 and 3-10, the expressions Q ′ β,0 (x) and Q " β,0 (x) in terms of Q β,1 (x) and Q β,2 (x). Equation 3-3 becomes

Q β,2 (x) + 4 (1 -x) Q β,1 (x) -4β (β -1) Q β,0 (x) = 0 (3-11)
From 3-11, by derivatives d dx and recurrence, it is easy to show that

Q β,k+2 (x) + 4 (k + 1 -x) Q β,k+1 (x) -4 [β (β -1) -k (k + 1)] Q β,k (x) = 0 (3-12)

C.2 CALOGERO N=3

We consider the Calogero differential operator

H Calogero = 3 i=1 d 2 dx 2 i + 2β i<j 1 x i -x j d dx i - d dx j (3-13)
and we look for solutions

H Calogero Φ (x i , y i ) = i y 2 i Φ (x i , y i ) (3-14)
where the solutions Φ (x i , y i )have a certain number of symmetry properties described somewhere else. We write

Φ (x i , y i ) = f (x i , y i ) e 3 i=1 x i y i (3-15)
so that the equations 3-13 and 3-14 become

D = 3 i=1 d 2 dx 2 i + 2β i<j 1 x i -x j d dx i - d dx j + y i -y j + 2 3 i=1 y i d dx i (3-16) and D f (x i , y i ) = 0 (3-17)
Let us introduce the variables

a = 1 2 (x 1 -x 2 ) (y 1 -y 2 ) = (x 1 -x 2 ) Y 12 (3-18) b = 1 2 (x 1 -x 3 ) (y 1 -y 3 ) = (x 1 -x 3 ) Y 13 (3-19) c = 1 2 (x 2 -x 3 ) (y 2 -y 3 ) = (x 2 -x 3 ) Y 23 (3-20)
where

Y ij = -Y ji (3-21)
and we look for solutions of the type f (a, b, c) . We have

df dx 1 = Y 12 f ′ a + Y 13 f ′ b (3-22) df dx 2 = Y 21 f ′ a + Y 23 f ′ c (3-23) df dx 3 = Y 31 f ′ b + Y 32 f ′ c (3-24)
and We further transform the result 3-39. We have

d 2 f dx 2 1 = Y 2 12 f " a 2 + 2Y 12 Y 13 f " ab + Y 2 13 f " b 2 (3-25) d 2 f dx 2 2 = Y 2 21 f " a 2 + 2Y 21 Y 23 f " ac + Y 2 23 f " c 2 (3-26) d 2 f dx 2 3 = Y 2 31 f " b 2 + 2Y 31 Y 32 f " bc + Y 2 32 f " c 2 (
Y 2 12 a = Y 12 (x 1 -x 2 ) = Y 12 ∆ (x) (x 1 -x 3 ) (x 2 -x 3 ) (3-40) Y 2 12 a = Y 12 ∆ (x) F -(x 1 -x 2 ) 2 (3-41)
where ∆ (x) = (x 1 -x 2 ) (x 1 -x 3 ) (x 2 -x 3 ) (3-42) where H Calogero is given in 3-13.

F = x 2 1 + x 2 2 + x 2 3 -x 1 x 2 -x 1 x 3 -

  n dO e Tr (X-xn)OY O -1 e xn Tr OY O -1 = e xn Tr Y G β,n dO e Tr (X-xn)OY O -1 (4-24)

5 - 21 )

 521 Let us introduce the vector e = (1, 1, . . . , 1) t(5-22)It is such that M is a stochastic matrix, i.e.:M.e = I β,n e , e t .M = I β,n e t (5-23)We thus have, for any polynomial P : e t P (K) e . I β,n = I β,n Tr P (Y )[START_REF] Carlitz | A Note on the Bessel Polynomials[END_REF][START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF][START_REF] Desrosiers | Duality in random matrix ensembles for all Beta[END_REF][START_REF] Duistermaat | [END_REF][START_REF] Dyson | Statistical Theory of the Energy Levels of Complex Systems. I[END_REF](10)[START_REF] Eynard | A short note about Morozovs formula[END_REF][START_REF] Eynard | Asymptotics of skew orthogonal polynomials[END_REF](13)(14)(15)(16)[START_REF] Kohler | Supersymmetric extensions of Calogero-Moser-Sutherlandlike models: construction and some solutions[END_REF](18)(19)(20)[START_REF] Jevicki | Collective field theory and Schwinger-Dyson equations in matrix models[END_REF][START_REF] Knapp | Representation theory of semisimple groups[END_REF][START_REF] Krall | A New Class of Orthogonal Polynomials: The Bessel Polynomials[END_REF][START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF] Notice that the Calogero equation is the caseP (K) = K 2 .If P is the characteristic polynomial of Y we get another differential equation for I β,n :

Theorem 6 . 1 (

 61 Conjecture of Brezin-Hikami):

3 - 27 )D 2 f 1 2 1 2 1 2

 3272111 The equations 3-16 and 3-17 becomeD f (a, b, c) = D 1 f (a, b, c) + D 2 f (a, b, c) = 0 (3-28) D 1 f (a, b, c) (a, b, c) = 2Y 12 Y 13 f " ab + the functions f (a, b, c) = Q β,k (a) a 2β Q β,k (b) b 2β Q β,k (c) c 2β = {k, k, k} (abc) 2β (3-31)where Q β,k (x) are defined in (3-4-3-5) and 3-6. We considerf (a) = Q β,k (a) β,k (a) -2β Q β,k (a) a 2β+1 = -2k Q β,k (a) + Q β,k+1 (a) Q β,k+2 (a) + (β + 2k + 2) Q β,k+1 (a) +2k (β + k + 1) Q β,k Q β,k+2 (a) + 2 (k + 1) Q β,k+1 (a) +2k (k + 1) Q β,k Q β,k+2 (a) + 2 (k + 1) Q β,k+1 (a) -2 [β (β -1) -k (k + 1)] Q β,k (a) (k+2 (a) + 4 (k + 1 -a) Q β,k+1 (a) -4 [β (β -1) -k (k + 1 -2a)] Q β,k (a)(3-37)Now, we use the recurrence relation 3-12 and get the simple result 2f "

x 2 x 3 ( 1 - 1 2 1 ( 1 2 2 -x 3 ) 2 - 1 8Γ∞ or β- 1 k=0ΓH

 3111123211 F -(x 3 -x 1 ) 2 (3-44) F -(x 2 -x 3 ) 2 (3-45)We note that in 3-39 the quantity F disappear sinceY 12 + Y 23 + Y 31 = x 2 ) a + circ.perm.] {k, f (a, b, c) = f (a) f (b) f (cbut in 3-30 the term β 2 ab f (a) f (b) f (c) disappears since Y 12 Y 13 ab + circ.perm. = x 2 -x 3 ∆ (x) + circ.perm. = 0 (3-51)Then, from 3-33 we get= 2Y 12 Y 13 [k Q β,k (a)+ 1 2 Q β,k+1 (a)] a 2β+1 [k Q β,k (b)+ 1 2 Q β,k+1 (b)] b 2β+1 Q β,k (c) c 2β +circ.perm. (3-52) Again, the term containing Q β,k (a) Q β,k (b) Q β,k (c) disappears by 3-51. We write 2β (x 2 -x 3 ) {k + 1, k + 1, k} +k {k + 1, k, k} + k {k, k + 1, k} +circ.perm. (3-53)We note that(x 2 -x 3 ) [{k + 1, k, k} + {k, k + 1, k} + {k, k, k + 1}] + circ.perm. abc) 2β (x 2 -x 3 ) {k + 1, k + 1, k} -k {k, k, k + 1} + circ.perm. (3-55)We now collect D 1 and D 2 . From 3-28, 3-31, 3-47 and 3-55 we obtainD {k, 2 -x 3 ) 4kc {k, k, k} + 1 2 {k + 1, k + 1, k} -k {k, k, k + 1} +circ.perm. {k, k, k} + circ.perm. 2 -x 3 ) 4k (c -k) {k, k, k} + 1 2 {k + 1, k + 1, k} -k {k, k, k + 1} +circ.perm. x 3 ) [ 1 2 {k + 1, k + 1, k} -4k (β -k) (β + k -1) {k, k, k -1} ] +circ.perm. k k! {k + 1, k + 1, k} -Γ(β-k+1) Γ(β+k-1) 1 8 k-1 (k-1)! {k, k, k -1} +circ.perm. (β -k) Γ (β + k) 1 8 k k! {k, k, k}(abc) 2β (3-64) so that D f (abc) = 0 + circ.perm. (3-65)We proved that a solution to 3-14 is Φ (x i , y i ) = Calogero Φ (x i , y i ) = 0 (3-67)

  n-1 x n y n+1 y n+2 . . . y 2n-1 y 2n +y 1 y 2 . . . y n-1 y n x n+1 x n+2 . . . x 2n-1 x 2n = (x 1 y n+1 + x n+1 y 1 )(x 2 . . . x n-1 x n y n+2 . . . y 2n-1 y 2n +y 2 . . .y n-1 y n x n+2 . . . x 2n-1 x 2n ) -x 2 . . . x n-1 x n x n+1 y n+2 . . . y 2n-1 y 2n y 1 -y 2 . . . y n-1 y n y n+1 x n+2 . . . x 2n-1 x 2n x 1 ≡ -x 2 . . . x n-1 x n x n+1 y n+2 . . . y 2n-1 y 2n y 1 -y 2 . . . y n-1 y n y n+1 x n+2 . . . x 2n-1 x 2n x 1 . . . x n-1 x n x n+1 y n+2 . . . y 2n-1 y 2n y 1 -y 2 . . . y n-1 y n y n+1 x n+2 . . . x 2n-1 x 2n x 1 ≡ -(x 2 y 1 + x 1 y 2 )(x 3 . . . x n-1 x n x n+1 y n+2 . . . y 2n-1 y 2n +y 3 . . . y n-1 y n y n+1 x n+2 . . . x 2n-1 x 2n ) +x n+1 x 1 x 3 . . . x n-1 x n y 2 y n+2 . . . y 2n + y n+1 y 1 y 3 . . . y n x 2 x n+2 . . . x 2n (1 -5)Repeating the same operation recursively we obtain ∀k:P n ≡ x n+1 . . . xn+k x 1 x k+2 . . . x n y 2 . . . y k+1 y k+n+1 . . . y 2n +y n+1 . . . y n+k y 1 y k+2 . . . y n x 2 . . . x k+1 x k+n+1 . . . x 2n (1 -6) In particular for k = n -2 we find: P n ≡ x n+1 . . . x 2n-2 x 1 x n y 2 . . . y n-1 y 2n-1 y 2n +y n+1 . . . y 2n-2 y 1 y n x 2 . . . x n-1 x 2n-1 x 2n ≡ (x 1 y 2n-1 + x 2n-1 y 1 )(x n+1 . . . x 2n-2 x n y 2 . . . y n-1 y 2n +y n+1 . . . y 2n-2 y n x 2 . . . x n-1 x 2n ) -x n . . . x 2n-1 y 1 y 2 . . . y n-1 y 2n -y n . . . y 2n-1 x 1 x 2 . . . x n-1 x 2n ) . . . x 2n-1 y 1 y 2 . . . y n-1 y 2n -y n . . . y 2n-1 x 1 x 2 . . . x n-1 x 2n ) ≡ -(x n y 1 + x 1 y n )(x n+1 . . . x 2n-1 y 2 . . . y n-1 y 2n +y n+1 . . . y 2n-1 x 2 . . . x n-1 x 2n ) +x 1 x n+1 . . . x 2n-1 y 2 . . . y n y 2n + y 1 y n+1 . . . y 2n-1 x 2 . . . x n x 2n (1 -8) n+1 . . . x 2n-1 y 2 . . . y n y 2n + y 1 y n+1 . . . y 2n-1 x 2 . . . x n x 2n ≡ (x 1 y 2n + x 2n y 1 )(x n+1 . . . x 2n-1 y 2 . . . y n + y n+1 . . . y 2n-1 x 2 . . . x n ) -x n+1 . . . x 2n y 1 y 2 . . . y n -y n+1 . . . y 2n x 1 . . . x n (1 -9)

	(1 -4) P n and we repeat the same operations: and then: ≡ P n ≡ -x n and once more -x 2 (1 -7) P n ≡ x 1 x

E-mail: michel.bergere@cea.fr

E-mail: bertrand.eynard@cea.fr

Acknowledgments

We would like to thank G. Akemann, E. Brezin, P. Desrosiers, S. Hikami, A. Prats-Ferrer, J.B. Zuber, for useful and fruitful discussions on this subject. This work is partly supported by the Enigma European network MRT-CT-2004-5652, by the ANR project Géométrie et intégrabilité en physique mathématique ANR-05-BLAN-0029-01, by the Enrage European network MRTN-CT-2004-005616, by the European Science Foundation through the Misgam program, by the French and Japaneese governments through PAI Sakurav, by the Quebec government with the FQRNT.

A Appendix:

Polynomials of τ Lemma A.1 Let P n (X, Y ) = x 1 x 2 . . . x n-1 x n y n+1 y n+2 . . . y 2n-1 y 2n +y 1 y 2 . . . y n-1 y n x n+1 x n+2 . . . x 2n-1 x 2n (1-1) We prove that P n is a polynomial of degree n in the τ 's, where τ i,2n+1 = -1 2 x i y i and τ i,j = -1 2 (x i -x j )(y i -y j ), with integer coefficients:

proof:

If α k = β l = 1, then we can apply lemmaA.1, and thus the lemma is proved for that case.

• therefore, we now assume that α k β l ≥ 2, and with no loss of generality we may assume that α k ≥ 2. We write:

From the recursion hypothesis, the RHS is a polynomial in the τ 's, and thus we have proved the lemma. Lemma A.3 Let P be a polynomial of 2n+2 variables x 1 , . . . , x n , x n+1 , y 1 , . . . , y n , y n+1 , with the following properties:

• P is invariant by translations ∀i, x i → x i + δx, y i → y i + δy,

• P is symmetric in the exchange X ↔ Y , Then P is a polynomial of the τ i,j 's.

proof:

Because of invariance by translation, we can always assume that x n+1 = y n+1 = 0. Then, the other properties imply that P is a linear combination of monomials of the type P α,β (X, Y ) of Lemma.A.2.

B Appendix: Loop equations

Loop equations for matrix models have been studied for a long time [27]. Loop equations (sometimes called Ward identities or Schwinger-Dyson equations) for β ensembles can be found for instance in [START_REF] Eynard | Asymptotics of skew orthogonal polynomials[END_REF][START_REF] Wiegmann | Large N expansion for the 2D Dyson gas[END_REF]19,20,[START_REF] Jevicki | Collective field theory and Schwinger-Dyson equations in matrix models[END_REF][START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF]. Here, we summarize the method.