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Orthogonal Polynomials for Potentials of two Variables with External Sources

This publication is an exercise which extends to two variables the Christoffel's construction of orthogonal polynomials for potentials of one variable with external sources. We generalize the construction to biorthogonal polynomials. We also introduce generalized Schur polynomials as a set of orthogonal, symmetric, non homogeneous polynomials of several variables, attached to Young tableaux.


Introduction
Recent progress in string theory [1] and in quantum chromodynamics [2] have reactualized the study of the spectral statistical properties of random matrices. Although most interesting results are expected from infinitely large matrices, exact results on the expectation values of ratios of random matrix determinants have been obtained for finite matrices, using the technique of orthogonal polynomials.
In this context, considerable results [9 − 10 − 11] collected recently in Ref. [3] can be found for potentials V (x) of one variable but relatively few and recent results [4] are known for potentials of two variables V (x, y) or equivalently V (z, z * ) where z = x + iy; however, such results could be of some physical interest, for instance, in the BMN program in string theory [5 − 6] .
It is the purpose of this publication to extend to potentials V (z, z * ) with external sources, the known results for potentials of one variable. Later on, as it is already done for potentials of one variable, we expect to generalize this structure to potentials V (z, z * ) with external sources both at the numerator and at the denominator, using the so called Cauchy transform.
This text is organized as follows: the various results are collected in Sect. 1, 2 and 3; the proofs are exposed in Sect. 4, 5, 6 and in the appendix. Finally, a discussion is proposed in Sect. 7 on orthogonal polynomials for potentials of one and two variables.
The construction of orthogonal polynomials for potentials of one variable with external sources is described as follows: given a real potential V (x) which admits an infinite set {p n (x)} of orthogonal monic polynomials (n is the degree in x of the polynomial) satisfying dx p m (x) p n (x) e −V (x) = h n δ nm (1) we consider L external sources (ξ 1 , ..., ξ L ) and we look for orthogonal monic polynomials q n (x; ξ i ) such that The solution to this problem is known since Christoffel [7] and reads p n (x) p n (ξ 1 ) ...
is needed to make the polynomial q n (x; ξ i ) monic.
The determinant in (3) The orthogonality of the polynomials p n (x) ensures the orthogonality of the polynomials q n (x; ξ i ) .
The pseudonorm k n (ξ i ) is found to be the ratio of two determinants Now we state a similar result for real potentials of two variables x and y.
We introduce the complex variables z = x + iy, z * = x − iy, and we consider a real potential V (z, z * ) which admits an infinite set {p n (z)} of orthogonal monic polynomials d 2 z p * m (z) p n (z) e −V (z,z * ) = h n δ nm (8) Everywhere in this text, f * (z) is a short notation for [f (z)] * . If the domain of integration and the potential are rotational invariant (V (zz * )), the set of orthogonal polynomials are sometimes called Ginibre's polynomials [8], namely p n (z) = z n (of course, the weights h n are different in each case); in general, the polynomials are not homogeneous and have complex coefficients.
We now consider L external complex sources (ξ 1 , ..., ξ L ) and we look for orthogonal monic polynomials q n (z; ξ i ; ξ * i ) such that |z − ξ i | 2 e −V (z,z * ) = k n (ξ i ; ξ * i ) δ nm (9) We show in Sect. 5 that the solution is given by the polynomials where the determinant is needed to make the polynomial q n (z; ξ i ; ξ * i ) monic.
The polynomials Q n (z; ξ * i ) are monic, of degree n in z and are defined from the property (12) and of course the corresponding complex conjugate properties. The proof of the orthogonality of the polynomials q n (z; ξ i ; ξ * i ) is exactly similar to the proof for a potential of one variable but here, of course, it relies upon the existence of the polynomials Q n (z; ξ * i ) satisfying (12). It is the purpose of Sect. 4 to show the existence, to construct and to give the properties of the polynomials Q n (z; ξ * i ) . For one external source ξ we show that In the general case with L external sources, we prove that the monic polynomials satisfy (12) and are unique. Clearly, all K n (z, ξ * i ) in (15) could be replaced by the corresponding set K q (z, ξ * i ) for any q satisfying n ≤ q < n + L.
We show in Sect. 5 that the polynomials q n (z; ξ i ; ξ * i ) may be written as q n (z; This result has been obtained by Akemann and Vernizzi in Ref. [4] (in their work K n (z, ξ * ) is called K n+1 z, ξ ).
The norm k n (ξ i ; ξ * i ) in (9) is proved to be as conjectured by Akemann and Vernizzi [4] who verified this formula up to L = 5. The results of this section are generalized below to biorthogonal polynomials and all proofs in Sect. 4 and 5 are performed in that case.

Biorthogonal Polynomials
Given two set of external sources η 1 , ..., η L1 and ξ 1 , ..., ξ L2 , we generalize the above constructions and define two sets of biorthogonal monic polynomials q n (z; ξ i ; η * i ) and q n (z; η i ; ξ * i ) satisfying the orthogonality relation For simplicity, we use the same symbol q n for the polynomials q n (z; ξ i ; η * i ) and q n (z; η i ; ξ * i ) although they are different functions depending of the number of variables η i and ξ i . The pseudonorms k n (ξ i ; η * i ) are complex and may vanish on some manifolds of the (ξ i ; η * i ) space. The uniqueness of the biorthogonal polynomials supposes that we consider generic values of the sources ξ i and η * i where the norms do not vanish.
The polynomials q n (z; ξ i ; η * i ) are naturally given by and correspondingly, the polynomials q n (z; η i ; ξ * i ) are obtained from the polynomials q n (z; ξ i ; η * i ) by exchanging the ξ's and the η's, an operation which implies the exchange of L 2 and L 1 . In (19), < n, L 2 > (ξ i ; η * i ) is defined as The polynomials Q n (z; η * i ) and Q n (z; ξ * i ) are respectively defined by the following properties: for m < n and the corresponding complex conjugate properties. Again, by simplicity, we use the same notation Q n in (21) and (22) although they are different functions depending of the number of variables η i and ξ i . The proof of the orthogonality of the polynomials q follows the corresponding proof for the potentials of one variable; the existence, the construction and the properties of the polynomials Q are given in Sect. 4. The polynomials q n (z; ξ i ; η * i ) take two different forms depending of the relative values of L 1 and and if L 1 ≥ L 2 + 1 where the determinant D n (ξ i ; η * i ) is defined for L 1 ≤ L 2 and becomes D * n (η i ; ξ * i ) for L 1 ≥ L 2 . In (23), Clearly, any line of K n can be replaced by the corresponding line of K n+q with 0 ≤ q ≤ L 2 − L 1 . We show in part 2 of the appendix that D n (ξ i ; η * i ) = 0 for n < L 1 − 1 and that Vandermonde determinant defined in (34). Also in (24), Again, any line of K n can be replaced by the corresponding line of K n+q with 0 ≤ q ≤ L 1 − L 2 . Similarly, we have D * n (η i ; ξ * i ) = 0 for n < L 2 − 1 and The pseudonorm k n (ξ i ; η * i ) is found to be If L 1 = L 2 we recover the result (17) with the ξ * j replaced by the η * j .
In Sect. 6 we apply the above formalism to the calculation of the integrals of the form where i and j run from 1 to N, a from 1 to L 1 and b from 1 to L 2 . These integrals are the generalization to potentials of two variables of similar integrals with potentials of one variable which have been expressed by Brezin and Hikami [9] as determinants of the corresponding orthogonal polynomials. The expressions (31-32) have been obtained before by Akemann and Vernizzi [4] . It is easily proved, from the property of the Vandermonde determinant (34) and from the biorthogonality of the polynomials q n (z; ξ i ; η * i ) , that This result can be transformed in terms of the determinants D n (ξ b ; η * a ) for and

A Generalization of the Schur Polynomials
We generalize the definition of the Schur polynomials of several variables (x 1 , ..., x n ) to a set of orthogonal, symmetric and non homogeneous polynomials of several variables attached to Young tableaux. We define where λ is a Young tableau described by the length of its rows λ 1 ≥ λ 2 ≥ ... ≥ λ n and ∆ (x i ) is the Vandermonde determinant corresponding to the empty Young tableau for any set of monic polynomials {π n (x)}. In the case where the polynomials p n (z) are Ginibre ′ s polynomials, the polynomials P λ (z i ) are Schur polynomials. The orthogonality of these polynomials is expressed as where the Young tableau λ ′ is obtained from the Young tableau λ by exchanging the rows and the columns, and λ ′ is the complementary Youg tableau of λ ′ in the rectangle [L × N ]; the surface of the Young tableau λ is denoted by |λ| .
Most of the results of the preceding sections can be expressed in terms of the generalized Schur polynomials. For instance where [L × n] is the Young tableau corresponding to the rectangle with L rows and n columns. Also the polynomials Q n (z; η * i ) are found to be where the Young tableau [L × n] j is the union of the rectangle [L − 1 × n] and of the smallest row λ L = j. Now, from (62-63) and part 2 of the appendix, we get for and for The polynomials q n (z; ξ i ; η * i ) can also be expressed as the ratio of two such expressions; for and for L 1 ≥ L 2 + 1 In (41) and (42) the denominators make the polynomials q n (z; and for Finally, we can also express the integrals (29) in terms of the generalized Schur polynomials; from (31-32) and part 2 of the appendix we obtain for L 1 ≥ L 2 and for L 1 ≤ L 2 In this section, we look for monic polynomials Q n (z; ξ * i ) satisfying the condition (12). We first show that the polynomials defined in (15) satisfy the condition (12); then, we prove that this solution is unique. First, we define the matrix elements which are zero for m > n. From (15) we obtain for m < n + L (48) Clearly, this determinant vanishes for n < m < n+L. However, it shows that for m < n + L the elements A n,m (ξ * i ) are linear combinations of the polynomials p * m (ξ i ) , i = 1, ..., L with L coefficients independant of m.
From the property (34) of the Vandermonde determinant we may write where r 1 , ..., r m are the roots of the polynomial p m (z) and ∆ r i , ξ j is the corresponding Vandermonde determinant. We now calculate the integral (12) for m < n; we find ∆ * r i , ξ j −1 times the determinant This determinant is zero for m < n since, for q < n + L, the elements A n,q (ξ * i ) are linear combinations of the polynomials p * q ξ j , j = 1, ..., L with the same q independant coefficients. This achieve the proof of (12) for the polynomials (15).
Before proving the unicity of the polynomials Q n (z; ξ * i ) , we prove that the integral This integral is calculated exactly as above with m replaced by n; since L > 0 we obtain ∆ * r i , ξ j −1 times the determinant This ends the proof of (51) .
We now prove the uniqueness of the polynomials Q n (z; ξ * i ) . The proof is a recurrence: Q 0 (z; ξ * i ) = 1; let us suppose that there exists two polynomials Of course, this is wrong for generic values of the variables ξ i .
We suppose now that we proved that the polynomials Q k (z; ξ * i ) are unique for k = 0, ..., n − 1 and consequently given by (15). If there exists two polynomials Q n (z; ξ * i ) and Q ′ n (z; ξ * i ) , then by difference, we have Because of the recurrence, the unique polynomial Q n−1 (z; ξ * i ) which satisfy (56) for k = 0, ..., n − 2 is given by (15); Consequently, for k = n − 1 equation (56) is equivalent to [n, L] * (ξ i ) = 0 which is wrong for generic values of the variables ξ i .
This ends the proof of the uniqueness of the polynomials Q n (z; ξ * i ) .
We now establish a first relation for the pseudonorm k n (ξ i ; η * i ) . From the definition (18) we easily obtain two relations These relations will be transformed further later on.
The rather lengthy calculation of < n, L 2 > (ξ i ; η * i ) is given in part 1 of the appendix; the expression of the result depends whether L 1 is larger or not to L 2 . We find where the determinants D n (ξ i ; η * i ) and D * n (η i ; ξ * i ) are defined in (25-26).

Applications
We exploit the existence of the biorthogonal polynomials to calculate the integrals (29). We write Then, we obtain (30) by developping the determinants and by integrating over each variables z i using the biorthogonality of the polynomials q n . Now, we establish the relation (36) between Then, we develop the determinant by separating the z i part from the ξ a part. We get Now, we define two Young tableaux λ and µ from the length of their rows We observe the relations We may now calculate I N,L2,L1 (ξ b ; η * a ) from the orthogonality property (35) of the polynomials P λ (z i ); we obtain where λ ′ 1 and λ ′ 2 are the complementary Young tableaux of λ ′ in the rectangles [L 1 × N ] and [L 2 × N ] respectively and where L = Inf (L 1 , L 2 ) . We thus obtain (45) or (46) by relabelling λ ′ 1 or λ ′ 2 as λ depending whether L 1 ≥ L 2 or not. Both formulas, (45 or 46) and (74), agree because of the following property proved in part 3 of the appendix: given a Young tableau λ in the rectangle [L × N ] and the corresponding Young tableau µ = λ ′ in the rectangle [N × L] , then

Discussion
Given a positive Borel measure µ (x) on the real line and its set of orthogonal, monic polynomials {p n (x)} satisfying we define two operations: Although µ 1 (x) and µ 2 (x) are not positive Borel measures, operations 1 and 2 transform the set of orthogonal polynomials p n (x) into new sets of orthogonal monic polynomials defined as -operation 1: -operation 2: where h n (y) is the Cauchy transform of p n (x) defined for y / ∈ R as h n (y) = 1 2iπ Of course, q 0 (x) = 1 and k 0 (y) = 2iπ h 0 (y) .
Successive iterations of operations 1 and 2 generate a set of orthogonal monic polynomials described by Uvarov [10] , namely, for a given set of external sources (ξ 1 , ..., ξ L ) at the numerator and (y 1 , ..., y M ) at the denominator, we have The orthogonal polynomials are, for n ≥ M  We note that in Uvarov's expression for (88), p q (y a ) for 0 ≤ q ≤ M − n− 1 is replaced by y q a (which is the same situation as in the Vandermonde determinant and leaves the value of the determinant unchanged).
As a consequence of this set of orthogonal polynomials, we can generalize the calculation of the integrals (84) to the integrals (91) this result has been obtained by Fyodorov and Strahov [11] ; for N ≤ M , we obtain where Similarly, for N ≤ M it is shown in part 4 of the appendix that and where the set of N variables θ a is any subset of (ξ 1 , ..., ξ M ) .If M = N there is no more p q (y j ) part in (95) and we recover(93). Now, if we take in (90) and (93) the derivatives of I N in regards to all variables ξ a , then, if we make all variables ξ a = y a , we prove in part 4 of the appendix that for N ≥ M and where " det " is a notation which means that we ignore the double poles at ξ a = ξ b as we develop the determinant. Moreover, it is easy to see that the residues of the single poles at ξ a = ξ b are zero, so that the integrals (97) are analytic in ξ's if the ξ's / ∈ R.
The discontinuity of h n (y) when y ∈ R is given by (83) disc h n (y) = p n (y) µ ′ (y) (100) where we use the notation dµ (y) = µ ′ (y) dy .Consequently, for N ≥ M the total discontinuity of J N , for a set of different real variables ξ ′ s (which is related to the N -point correlation function), is (101) as already obtained in Ref. [12 − 13] . For N < M the expression for J N is more complicate but the total discontinuity in that case is clearly zero.
A similar program should be achieved for potentials of two variables V (z, z * ).
We consider a positive Borel measure µ (z, z * ) on the complex plane and its set of orthogonal, monic polynomials {p n (z)} satisfying We define four operations: Operations 1 and 2 transform the set of orthogonal polynomials p n (z) into sets of biorthogonal monic polynomials defined as -operation 1: In (107-108), Φ means an empty set of variables, Q n and K n are defined in (13)(14) and have the property that Q * n (z; ξ * ) p m (z) (z − ξ) dµ (z, z * ) = 0 for m < n (110) -operation 2: where, similarly to (110), we have Successive iterations of operations 1 and 2 is the purpose of this publication and generate the biorthogonal polynomials q n (z; ξ i ; η * i ) and q * n (z; η i ; ξ * i ) described in (18-19). Operations 3 and 4 are beyond the scope of this publication and might be described in a near future.
where we sum over all indices {j 1 , ..., j L2 } running from 1 to L 1 . In (A6), the antisymmetry of the determinant in regards to the exchange of the variables η * ji is such that we may reconstruct a determinant from the sum of products In 3 • ) of part 1 of the appendix, we prove that for 1 ≤ j 1 < ... < j L2 ≤ L 1 where the notation [n + L 2 , L 1 − L 2 ] * ( j1,..., jL 2 ) (η i ) means that the columns relative to the variables η * j1 , ..., η * jL 2 have been ommitted in the determinant. Consequently, . From the definition (26), the above sum is nothing but D * n+L2−1 (η i ; ξ * i ) , developped relatively to the bloc K n+L2−1 by choosing the columns {j 1 < ... < j L2 } and summing over all possible permutations. This achieve the proof of (63). We introduce (L 1 − L 2 ) independant functions N i (z) for i = 1, ..., L 1 − L 2 and we consider the sum where we sum over all indices {α 1 , ..., α L1 } running from 1 to L 1 .
On one hand, a direct calculation shows that J is the determinant The question marks ? are easy to compute but useless and the zeroes come from the fact that we reconstruct determinants with two lines alike. The L 2 bottom lines of the determinant is a [L 2 × L 1 − L 2 ] rectangle of zeroes and a [L 2 × L 2 ] triangular matrix with zeroes below the diagonal and ? above. Consequently On the other hand, we may use the antisymmetry in the indices α i of the determinant in (A10) to reconstruct a determinant with the p ′ s which is nothing but [n, L 1 ] (η i ) ; the quantity J is now obtained from a restricted sum 1≤α1<...<αL 1 ≤L1 and such a restriction implies α i = i. We may write J as Then, given a set {j k } of L 2 indices such that 1 ≤ j 1 < ... < j L2 ≤ L 1 , we define {s p } as the set of (L 1 − L 2 ) indices complementary of {j k } in {1, ..., L 1 } and such that 1 ≤ s 1 < ... < s L1−L2 ≤ L 1 . Now, we look in both expressions (A13 and A14) of J for the coefficient of N L1−L2 η s1 ...N 1 η sL 1 −L 2 ; in the expression (A14) we obtain In the expression (A13) the corresponding coefficient is found to be If we compare (A15) and (A16) we obtain (A8) up to complex conjugaison.

Part 2:
We calculate D n (ξ i ; η * i ) and D * n (η i ; ξ * i ) defined in (25-26) in terms of the generalized Schur polynomials (33). From (25) and (14) we have in the case ... p α L 1 ξ L2 (A17) where we sum over L 1 indices α k such that 0 ≤ α k ≤ n. Because of the antisymmetry in the indices α k we can reorganize the sum into a restricted sum and we introduce a second determinant Clearly, if n < L 1 − 1, there is not enough room for the indices α k and D n (ξ i ; η * i ) = 0; if n = L 1 − 1, the indices are fixed to α k = L 1 − k so that . Now, for n ≥ L 1 − 1 we introduce the generalized Schur polynomials defined in (33) and we write for where λ is a Young tableau in the rectangle [L 1 × (n − L 1 + 1)] with the rows of length λ k = α k − L 1 + k.
A similar demonstration gives for L 1 ≥ L 2 and n ≥ L 2 − 1 Part 3: We prove the relation (75). Given a Young tableau λ in the rectangle [L × N ] and the corresponding Young tableau µ = λ ′ (as defined in Sect. 3 (36)) in the rectangle [N × L] we establish a relation between the lengths of the rows of λ and the lengths of the rows of µ. The lengths of the rows of λ are defined by L numbers satisfying N ≥ λ 1 ≥ ... ≥ λ L ≥ 0. We define the numbers ν k as the numbers of rows of λ with length k. We have where |λ| is the surface of the Young tableau λ. Finally, we define so that σ N = L; the sequence {σ k } is a weakly increasing sequence as k runs from 0 to N. We have the following properties: for 0 ≤ k ≤ N, if ν k > 0, then with σ −1 = 0 by convention. Now, we define the numbers The sequence {α i } is a strictly decreasing sequence as i runs from 1 to L. Because of (A24), If ν k > 0 α L−σ k +1 to α L−σ k−1 are consecutive numbers from σ k + k − 1 to σ k−1 + k (A26) Clearly, the numbers α i are all numbers from N + L − 1 to 0 at the exception of the set {σ k + k} for k = 0, ..., N − 1. We note that if ν k = 0 with ν k+1 ν k−1 > 0, we get two consecutive missing numbers σ k + k and σ k−1 + k − 1 = σ k + k − 1; if p consecutive ν ′ s are nul, we get (p + 1) consecutive numbers missing in the set {α i } and belonging to the set {σ k + k} . The set {σ k + k} is a strictly increasing sequence as k runs from 0 to N − 1. We may write We now consider the Young tableau λ ′ in the rectangle [N × L] and symmetric of λ with regards to the diagonal line. It is easy to convince oneself that and consequently, if µ = λ ′ the lengths of the rows of µ are The sequence {µ k } is a weakly decreasing sequence as k runs from 1 to N. We define The sequence {β k }is a strictly decreasing sequence as k runs from 1 to N.
We just proved that and this concludes the proof of (75).

Part 4:
This part of the appendix is devoted to the proof of the equations (93-95) and (97). 1 • ) We first mention a generalisation of Christoffel-Darboux relation. Given two sets of K variables (ξ 1 ...ξ K ) and (η 1 ...η K ) , from (90-91) we may evaluate the following integrals as On the other hand, we may write and similarly for the η b part, we may introduce the Vandermonde determinants of the orthogonal polynomials p n (x) and integrate; we get The sum in (A35) is nothing but det We just proved that For K = 1, equation (A37) is nothing but Christoffel-Darboux relation Also, for N = 0 we have We wish to evaluate the integrals Clearly, where H n (θ, y) is defined in (94) in terms of the Cauchy transform. Also, We observe the recurrence relation f L (ξ 1 , ..., ξ L , y, θ) = (y − ξ L ) f L−1 ξ 1 , ..., ξ L−1 , y, θ + const(ξ 1 , ..., ξ L−1 , θ) (A43) where constant means y independent. It is easy to see that the constant in (A43) is nul if the two following conditions are fulfilled: first, n ≥ L − 1, second θ = ξ b for b = 1, ..., L − 1. Of course, f L (ξ 1 , ..., ξ L , y, θ) is completely symmetric in the ξ's and the same result is valid for θ = ξ L by simply performing the recurrence (A43) from y − ξ b =L . Consequently, n (ξ a , y) = L a=1 (y − ξ a ) 2iπ H n (ξ b ; y) + 1 y − ξ b for b = 1, ..., L ≤ n + 1 (A44) Finally, using the same recurrence (A43) again, we may write for b = 1, ..., L > n + 1 n (ξ a , y) = For each column of the determinant, we can perform the integration by applying dµ