The large-N limit of matrix integrals over the orthogonal group

Jean-Bernard Zuber

LPTHE (CNRS, UMR 7589), Université Pierre et Marie Curie-Paris 6, 75252 Paris Cedex, France

E-mail: jean-bernard.zuber@upmc.fr

Received 7 July 2008, in final form 17 July 2008
Published 21 August 2008

1. Introduction

Matrix integrals of the type

over a classical compact group G =  U(N), O(N) or Sp(N), with κ being a real parameter, appear frequently in theoretical physics, from disordered systems [1–3] to 2D quantum gravity [4] and related topics. They also have a mathematical interest in connection with integrability, statistics and free probabilities. They are sometimes called matrix Bessel functions [5], or (generalized) HCIZ integrals. While the expression of Z is well known for the group U(N) [6, 7], the situation with O(N) is more subtle. The result is known for skew-symmetric matrices A and B [6], but its form is only partially understood for the more frequently encountered case of symmetric matrices, in spite of recent progress [5, 8, 9].

On the other hand, in the large-N limit, we expect things to simplify [10, 11]. It is the purpose of this work to revisit this old problem and to show that log Z has universality properties in the large-N limit, a pattern which does not seem to have been stressed enough before, at least in the physics literature (see the historical note below).

This paper is organized as follows. In section 2, we discuss the related but simpler case of the integral `in an external field',

(where the second term in the exponential will be omitted in the orthogonal case). We prove that it enjoys a universality property in the large-N limit. In section 3, we turn to integral (1), discuss its large-N limit and prove that it has a similar universality property. We also extend our integrals to cases where matrices A and B are neither symmetric (or Hermitian) nor antisymmetric.

For the sake of simplicity this paper will be focused on the case of G =  O(N) as compared to U(N), but similar considerations apply to Sp(N).

2. The integral (2)

2.1.  The basic integrals and their generating function

Let us consider the basic integral over the orthogonal group O(N)

where O is an N × N orthogonal matrix and i, respectively j, are n-tuples of indices i respectively j.

One may easily prove that is non-vanishing only for even n and has then the following general structure [19]:

where P_2n denotes the set of pairings in {1, 2,  ..., 2n},

and where the coefficients C(p₁, p₂) enjoy many properties and may be determined recursively, see appendix A for a review.

The analogous basic integral for U(N) is

with a double sum over permutations σ, τ   S_n, the symmetric group, see appendix A.

Another way of encoding these formulae is to use the generating functions

where J is a generic (i.e. non-symmetric, respectively non-Hermitian) matrix. depends only on traces of powers of J · J^t, again by invariance of the integral under J → O₁ · J · O₂. Likewise, depends only on J · J†.

2.2.  The large-N limit and its relation to the unitary case

In the large-N limit, one may show that

exists and is related to the corresponding expression for the unitary group.

Let us recall the situation in the unitary case. The generating function has been studied extensively in the past. The limit was shown to satisfy a partial differential equation with respect to the eigenvalues of J · J† [11]. In the `strong coupling phase', an explicit expression was given [12] for the expansion of W_U in a series expansion in traces of powers of J · J†

where αn denotes a partition of n  =  α₁ · 1  +  α₂ · 2  +  · · ·  +  α_n · n and

Now we claim that W_O defined in (9) is

Proof. We repeat the steps of [11], paying due attention to the differences between independant matrix elements in a complex Hermitian and in a real symmetric matrix. The trivial identity is re-expressed in terms of the eigenvalues λ_i of the real symmetric matrix J · J^t:

Writing as above and dropping subdominant terms in the large-N limit, with W_O and W_i := N∂W_O/∂λ_i of order 1, we obtain

which is precisely the equation satisfied by in [11]. This, supplemented by appropriate boundary conditions, suffices to complete the proof of (13).        

As noted elsewhere [7, 13] it is appropriate to expand W on `free' (or `non-crossing') cumulants [7, 14, 15]

Then

where the only occurrence of ψ₁ is in the first term.

3. The generalized HCIZ integral

3.1.  Notations. Review of known results

Let us first recall the well-known results. If the matrices A and B in (1) are in the Lie algebra of the group G, namely are real skew-symmetric, respectively anti-Hermitian, for G  =  O(N), respectively U(N), the exact expression of Z^(G) is known from the work of Harish-Chandra [6]. To make the formulae quite explicit, we take A and B in the Cartan torus, i.e. of a diagonal or block-diagonal form [16]

and

and likewise for B. We assume that all the a's are distinct and likewise for the b's. Then the integral (1) reads

where

For convenience, Δ_U(a) will be abbreviated into Δ(A)  =  ∏_i<j(a_i – a_j), the usual Vandermonde determinant.

3.2.  The large-N limit

We now consider

Note the factor 2 introduced for convenience in the latter exponential. We claim that, for A and B both symmetric or skew-symmetric,

3.2.1.  The skew-symmetric case

For A and B both skew-symmetric, the limit of (20) is easy to evaluate. Assuming without loss of generality that all a's and b's are positive, we find that , thus . On the other hand, the real matrices A and B of the form (19) may also be regarded as anti-Hermitian, with eigenvalues A_j  =  ±ia_j, j  =  1,  ..., m (supplemented by 0 if N  =  2m  +  1). An easy computation gives Δ(A)  =  (Δ_O(a))² up to a sign. The matrix , as N → ∞. Hence . Thus, the U(N) integration for the pair (A, B) yields, according to (21), and up to an overall factor,

in accordance with (24).

3.2.2.  The symmetric case

We now turn to the more challenging case where both A and B are real symmetric. We may suppose that A and B are in a diagonal form A  =  diag(a_i), B  =  diag(b_i), and assume that all a's and all b's are distinct. In that case, we shall resort to (an infinite set of) differential equations, in a way similar to the discussion of section 2.

In a recent work, Bergère and Eynard [9] have introduced the following integrals over the compact group G:

which may be regarded as particular two-point correlation functions associated with the partition function Z^(G) of (1). The latter is recovered from M_ij by a summation over i or j

As shown in [9], M_ij satisfy the following set of differential equations:

with no summation over k in the rhs. Here K refers to the matrix differential operator

(We make use of Dyson's label β  =  1, 2 for G  =  O(N), U(N), respectively.)

Now, by a repeated application of the operator K on M, we find for any positive integer p that ∑_j(K^p)_ij M_jk  =  N^pM_ikb^p_k; hence after summation over i and k,

The differential operator D_p := ∑_1≤i,j≤N(K^p)_ij has thus the property that

Thus far, the discussion holds for any finite value of N. Now take the large-N limit with the ansatz . Equation (31) reduces in that limit to

with of order 1, as in section 2, and all other terms resulting from further application of over F^(G) suppressed by inverse powers of N.

These equations had been obtained in [7] in the case of U(N) (β  =  2) from the explicit form of Z^(U) and shown to determine recursively the expansion of F^(U) in traces of powers of A and B. Comparing the orthogonal (κ  =  β  =  1) and unitary (κ  =  β  =  2) cases, it is clear in (32) that 2F^(O)(A, B) satisfies the same set of equations as F^(U)(A, B), thus vindicating (24).

3.3.  The generic case

Although it does not look very natural in view of their symmetries, one may extend the integrals Z^(U) and Z^(O) to the case of generic complex (non-Hermitian), respectively real (neither symmetric nor skew-symmetric), matrices A and B. If we insist on having real quantities in the exponential, the unitary integral that we consider reads

(and one recovers the factor 2 introduced in (23) for A and B Hermitian). In parallel the more general orthogonal integral reads

The functions Z^(U) and F^(U) have now expansions in traces of products of A and A† (or A^t) and of B and B† (or B^t), with an equal number of daggers (respectively transpositions) appearing on A and B. We can no longer rely on the diagonal form of A and B (a generic real, respectively complex matrix is not diagonalized by an orthogonal, respectively unitary, matrix) and there are no longer differential equations in these eigenvalues satisfied by Z or F. Still, there is some evidence that universality holds again. By expanding the exponentials and by making use of the explicit integrals (4), (6), (see also appendix A), we have checked that, up to the fourth order, for A and B real

If we write the expansion of F in powers of A (and B) as F  =  ∑_n = 1 F_n, we find

where the ψ_n are the free cumulants defined above and the ψ_nt are `polarized' versions of the latter, involving A and A† (or A^t), see appendix B for explicit expressions.

4. Concluding remarks

• It should be stressed that the equality (24) can be true only asymptotically as N → ∞. Indeed the exact result [6] for A and B skew-symmetric as well as what is known for A and B both symmetric [8] clearly indicate that it does not hold for finite N.
• The differential operator D_p considered above is interesting in its own right. Consider the differential operator such that

  

  For Hermitian matrices, for which all matrix elements A_ij may be regarded as independent, one may write

  

  while in the case of symmetric matrices, the general expression involves some combinatorial factors. The above property (40) suffices to define on any (differentiable) function of A, by the Fourier transform.

  Now let act on functions f(A) invariant upon A → ΩAΩ^–1. Then reduces to a differential operator D_p on the eigenvalues a_i of A. As we have seen above, D_p  =  ∑_ij(K^p)_ij, but it would seem desirable to have a more direct construction of that basic operator. In the case of G  =  U(N), one has the elegant form [7]

  

  This result, however, makes use of the explicit form (20) of Z^(U), and there is no counterpart for G  =  O(N). Thus the question is: can one derive the expression (42) of D_p from that (41) of ? Curiously, what looks like an innocent exercise of calculus turns out to be non-trivial, even for G  =  U(N).

• In view of the similarity between (13) and (24), on one hand, and of our (partial) results and conjecture on the `generic' case, on the other, it would be nice to have a general, intuitive argument why these universality properties hold. Heuristically, the overall factor in (13) and (24) just reflects the ratio of numbers of degrees of freedom in the two cases: there are N(N – 1)/2 ~ N<sup>2</sup>/2 real parameters in an orthogonal matrix and N<sup>2</sup> in a unitary one. But why is the function of A and B universal?
• Diagrammatics? A diagrammatic expansion exists for F<sup>(U)</sup> [13], using the functional of section 2, and this matches a combinatorial expansion [17]. Repeating the argument in the real orthogonal case leads to a much less transparent result, however, and does not seem to yield a simple derivation of (24) based on (13).
• A heuristic argument. Our result (24) for real symmetric versus complex Hermitian matrices should also be related to a similar relation between partition functions of two-matrix models. It is `well known' that integrals over two real symmetric, respectively two complex Hermitian matrices with arbitrary polynomial potentials V and W,

  

  

  are such that for large N

  

  (Note once again the factor 2 in front of the `action' of the Hermitian case.)

  For one-matrix integrals, this is a classical result, following from the saddle point approximation [14] or from the orthogonal polynomial approach [18]. It may also be derived from the diagrammatics: Feynman diagrams for real symmetric matrices in the large-N limit are the same as those of the Hermitian integral, up to factors of 2 coming from the possible twists of the double lines of their propagators. This diagrammatic argument is expected to extend to the two-matrix integrals (44), justifying the claim (45).

  On the other hand, if we diagonalize the matrices A  =  diag(a_i) and B  =  diag(b_i), we see that (44) reduces to

  

  

  Finally, if we imagine that the latter integrals over the eigenvalues are dominated in the large-N limit by a saddle point configuration, we see that the scaling (24) of the angular part is consistent with the scaling (45) of the full integral. Obviously a more rigorous version of this heuristic argument would be desirable.

• Historical remarks. As far as we know, the property (13) had never been observed before. On the other hand, property (24) has a richer history. It seems to have been first observed in the case where A or B is of finite rank in [2], and then repeatedly used in the physics literature [22, 21]. This was later proved in a rigorous way in [17]. In [20], this is extended to the case where the rank is o(N). Indeed for a finite rank of A, say, only terms with a single trace of some power of A dominate, and the expression of F(A, B) is known to be given by for the unitary group [7].

  Following a totally different approach, Guionnet and Zeitouni [23] have proved rigorously the existence of the free energies F^(U) and F^(O) (for A and B symmetric) in the large limit, and have established that they solve the flow equation proposed by Matytsin [24]. A by-product of their discussion is the explicit β dependence of the free energy and the resulting universality property (24). This has been made more explicit in the recent paper [25]. These papers also cover the case of the symplectic group (β  =  4).

Acknowledgments

The author was supported by `ENIGMA' MRT-CT-2004-5652, ESF program `MISGAM' and ANR program `GIMP' ANR-05-BLAN-0029-01. He wants to thank A Guionnet and P Zinn-Justin for discussions, and Y Kabashima for revigorating his interest in these integrals. Special thanks go to M Bergère and B Eynard for communicating their results prior to publication.

Appendix A. More details on the `basic integrals'

In this appendix, we recall well-known results [10] on the integral (3). Equivalently we may consider

where u_a and v_a, a  =  1,  ..., n, are vectors of . The integral is linear in each u_a and each v_a, and is invariant under a global rotation of all u's or of all v's: u_a → O₁ u_a, v_a → O₂ v_a, since this may be absorbed by the change of integration variable O^t₁ OO₂ → O in accordance with the invariance of the Haar measure DO. If N > n the completely antisymmetric tensor cannot be used to build invariants. Hence is only a function of the invariants u_a. u_b, v_a. v_b and by linearity must be of the form

a sum over all possible pairings of the indices a  =  1,  ..., n, b  =  1,  ..., n; this shows that vanishes for n odd. In the following we change n → 2n and denote P_2n the set of all pairings of {1, 2,  ..., 2n}, with |P_2n|  =  (2n – 1)!!.

Then the general expression of is indeed of the form (4). The coefficients C(p₁, p₂) may be determined recursively, but let us first point some general features.

Proof. To any cycle α of p₁.p₂, {a, p₁.p₂(a), (p₁.p₂)²(a),  ..., (p₁.p₂)^r(a)}, one may associate another one {p₁(a), p₁.p₂.p₁(a), (p₁.p₂)²p₁(a),  ..., (p₁.p₂)^rp₁(a)}, which is obviously of the same length and which acts on distinct elements. Thus p₁.p₂  =  σ.σ ', where σ and σ ' acting on distinct elements of {1, 2,  ..., 2n} may be regarded as in the same class of S_n. Moreover the class [p₁ · p₂], i.e. the cycle structure of p₁ · p₂ is obviously given by that of [σ]  =  [σ '].        

The coefficients C are then determined recursively. Noting that by contracting the last two j indices one constructs , and one gets a (strongly overdetermined) system of equations relating the C's of order n to those of order n – 1 [10]. Explicit although fairly complicated solutions have been given [19].

The first coefficients read explicitly

The analogous basic integrals in U(N) are more widely known, see (6). One may actually give an explicit form to the C([σ.τ]), namely

where χ^(λ)([σ]) is the character of the symmetric group S_n associated with the Young diagram Y, hence a function of the class [σ] of σ; χ^(λ)(1) is thus the dimension of that representation; s_λ(X) is the character of the linear group GL(N) associated with the Young diagram Y, i.e. a Schur function when expressed in terms of the eigenvalues of X; s_λ(I) is thus the dimension of that representation.

The first coefficients read explicitly

Appendix B. Free (non-crossing) cumulants

Note that in this appendix, we make use of a different notation for normalized traces the usual trace, thus tr I  =  1.

For convenience, we list here the first free cumulants of A in terms of the together with the mixed ones, involving traces of products A and A†:

[1] 

Kazakov V A 1986 Phys. Lett. A 119 140–4 
CrossRef

[2] 

Marinari M, Parisi G and Ritort F 1994 J. Phys. A: Math. Gen. 27 7647–68 
IOPscience

[3] 

Guhr T, Müller-Groeling A and Weidenmëller H A 1998 Phys. Rep. 299 189–428 (Preprint cond-mat/9707301) 
CrossRefPreprint

[4] 

Di Francesco P, Ginsparg P and Zinn-Justin J 1995 Phys. Rep. 254 1–133 (Preprint hep-th/9306153) 
CrossRefPreprint

[5] 

Guhr T and Kohler H 2002 J. Math. Phys. 43 2707 (Preprint math-ph/0011007) 
CrossRefPreprint

[6] 

Harish-Chandra  1957 Am. J. Math. 79 87–120 
CrossRef

[7] 

Itzykson C and Zuber J-B 1980 J. Math. Phys. 21 411–21 (erratum on http://www.lpthe.jussieu.fr/~zuber) 
CrossRef

[8] 

Brézin É and Hikami S 2003 Commun. Math. Phys. 235 125–37 (Preprint math-ph/0208002) 
CrossRefPreprint
Brézin É and Hikami S 2006 WKB-expansion of the Harish-Chandra–Itzykson–Zuber integral for arbitrary beta Preprint math-ph/0604041  
Preprint

[9] 

Bergère M and Eynard B 2008 Some properties of angular integrals Preprint 0805.4482  
Preprint

[10] 

Weingarten D 1978 J. Math. Phys. 19 999–1001 
CrossRef

[11] 

Brézin É and Gross D 1980 Phys. Lett. B 97 120–4 
CrossRef

[12] 

O' Brien K and Zuber J-B 1984 Phys. Lett. B 144 407–8 
CrossRef

[13] 

Zinn-Justin P and Zuber J-B 2003 J. Phys. A: Math. Gen. 36 3173–93 (Preprint math-ph/0209019) 
IOPsciencePreprint

[14] 

Brézin É, Itzykson C, Parisi G and Zuber J-B 1978 Commun. Math. Phys. 59 35–51 
CrossRef

[15] 

Speicher R 1994 Math. Ann. 298 611–28 
CrossRef

[16] 

Prats-Ferrer A, Eynard B, Di Francesco P and Zuber J-B 2007 J. Stat. Phys. 129 885–935 (Preprint math-ph/0610049) (erratum on http://www.lpthe.jussieu.fr/~zuber) 
CrossRefPreprint

[17] 

Collins B PhD 2003 Int. Math. Res. Not. 17 953–82 (Preprint math-ph/0205010) 
CrossRefPreprint

[18] 

Brézin É and Neuberger H 1991 Nucl. Phys. B 350 513–53 
CrossRef

[19] 

Collins B and Śniady P 2006 Commun. Math. Phys. 264 773–95 (Preprint math-ph/0402073) 
CrossRefPreprint

[20] 

Collins C and Śniady P 2005 New scaling of Itzykson–Zuber integrals Preprint math/0505664  
Preprint

[21] 

Gallucio S, Bouchaud J-P and Potters M 1998 Preprint cond-mat/9801209  
Preprint

[22] 

Cherrier R, Dean D S and Lefèvre A 2003 Phys. Rev. E 67 046112 (Preprint cond-mat/0211695) 
CrossRefPreprint

[23] 

Guionnet A and Zeitouni O 2002 J. Funct. Anal. 188 461–515 
CrossRef

[24] 

Matytsin A 1994 Nucl. Phys. B 411 805–20 (Preprint hep-th/9306077) 
CrossRefPreprint

[25] 

Collins B, Guionnet A and Maurel-Segala E 2002 Asymptotics of unitary matrix integrals, May 2008 preprint