A short note about Morozov's formula

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A short note about Morozov’s formula

Bertrand Eynard

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ccsd-00002150, version 1 - 25 Jun 2004

SPhT-T04/077

A short note about Morozov’s formula

B. Eynard1

Service de Physique Th´eorique de Saclay, F-91191 Gif-sur-Yvette Cedex, France.

Abstract

The purpose of this short note, is to rewrite Morozov’s formula for correlation functions over the unitary group, in a much simpler form, involving the computation of a single determinant.

1 Introduction

The main result of this paper is given in 2.13.

In [1], A. Morozov proposed a formula for correlation functions of unitary matrices with Itzykson-Zuber’s type measure:

h|Uji|2iU(N ) := 1 I(X, Y ) Z U(N ) dU|Uji|2etr (XU †_{Y U)} (1.1) where dU is the Haar measure over the unitary group U(N) (appropriately normalized, so that 1.4 below holds with prefactor 1), and X and Y are two given diagonal complex matrices:

X = diag(x1, . . . , xN) , Y = diag(y1, . . . , yN) (1.2)

and the normalization factor I(X, Y ) is the so-called Harish-Chandra-Itzykson-Zuber integral: I(X, Y ) := Z U(N ) dUetr (XU†Y U) (1.3) 1_{E-mail: eynard@spht.saclay.cea.fr}

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It is well-known [4, 5] that:

I(X, Y ) = det E

∆(X)∆(Y ) (1.4)

where E is the matrix with entries:

Eij := exiyj (1.5)

and where ∆(X) and ∆(Y ) are the Vandermonde determinants: ∆(X) :=Y i<j (xi− xj) , ∆(Y ) := Y i<j (yi− yj) . (1.6)

Here we shall assume that det E 6= 0 and ∆(X) 6= 0 and ∆(Y ) 6= 0.

Morozov’s formula was proven in [2], from Shatashvili’s formula [3]. Morozov’s for-mula was originaly written as follows, for any arbitrary sequences of complex numbers ai, bj one has: N X i,j=1 aibj Z U(N ) dU|Uji|2etr (XU †_{Y U)} = 1 ∆(X)∆(Y ) X ρ∈SN ǫ(ρ)eP xℓyρ(ℓ) N −1 X n=0 (−1)n X i1<i2<...<in+1 det       1 · · · 1 xi1 · · · xin+1 ... ... xn−1_i 1 · · · x n−1 in+1 ai1 · · · ain+1       det     1 · · · 1 xi1 · · · xin+1 ... ... xn i1 · · · x n in+1     det       1 · · · 1 yρ(i1) · · · yρ(in+1) ... ... y_ρ(in−1 1) · · · y n−1 ρ(in+1) b_ρ(i₁₎ · · · bρ(in+1)       det     1 · · · 1 yρ(i1) · · · yρ(in+1) ... ... yn ρ(i1) · · · y n ρ(in+1)     , (1.7)

where SN is the smmetric group of rank N.

2 Rewriting Morozov’s formula

The purpose of this short note is to rewrite this formula in a much simpler form, in a way very similar to what was done in [2].

For any two complex numbers x and y (such that |x| > max |xi| and |y| > max |yi|),

consider the choice: ai = 1 x− xi = ∞ X r=0 xr i xr+1 , bi = 1 y− yi = ∞ X s=0 ys i ys+1 (2.8)

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Introduce the Schur polynomials (corresponding to hook diagramms): Sr(xi1, . . . , xin+1) := det       1 · · · 1 xi1 · · · xin+1 ... ... xn−1_i₁ · · · xn−1in+1 xr i1 · · · x r in+1       det     1 · · · 1 xi1 · · · xin+1 ... ... xn i1 · · · x n in+1     = X a1≤a2≤...≤ar−n r−n Y k=1 xi_ak = X j1+···+jn+1=r−n xj1 i1 · · · x jn+1 in+1 . (2.9)

The formal generating function of these Schur polynomials is:

∞ X r=0 1 xr+1Sr(xi1, . . . , xin+1) = n+1 Y k=1 1 x− xik (2.10) Inserting that into 1.7, one gets:

I(X, Y ) X i,j 1 x− xi 1 y− yj h|Uji|2iU(N ) = Z U(N ) dU tr 1 x− XU 1 y− Y U † etr (XU†_{Y U)} = 1 ∆(X)∆(Y ) ∞ X r,s=0 X ρ∈SN ǫ(ρ)eP xℓyρ(ℓ) N −1 X n=0 (−1)nX i1<i2<...<in+1 Sr(xi1, . . . , xin+1) xr+1 Ss(yρ(i1), . . . , yρ(in+1)) ys+1 = 1 ∆(X)∆(Y ) X ρ∈SN ǫ(ρ)eP xℓyρ(ℓ) N −1 X n=0 (−1)nX i1<i2<...<in+1 n+1 Y k=1 1 x− xik n+1 Y l=1 1 y− yρ(il) = 1 ∆(X)∆(Y ) X ρ∈SN ǫ(ρ)eP xℓyρ(ℓ) " 1 − N Y i=1 1 − 1 x− xi 1 y− yρ(i) # = 1 ∆(X)∆(Y ) X ρ∈SN ǫ(ρ) " _N Y i=1 exiyρ(i) ₋ N Y i=1 exiyρ(i) ₋ 1 x− xi exiyρ(i) 1 y− yρ(i) # = 1 ∆(X)∆(Y ) det ( exiyj_{) − det} exiyj ₋ 1 x− xi exiyj 1 y− yj (2.11)

Thus, Morozov’s formula can be rewritten: Z U(N ) dU tr 1 x− XU 1 y− Y U † etr (XU†_{Y U)} = det E − det E−_x−X1 E_y−Y1 ∆(X)∆(Y ) (2.12)

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or: tr 1 x− XU 1 y− Y U † = 1 − detE−_x−X1 E_y−Y1 det E = 1 − det 1 − 1 x− XE 1 y− YE −1 (2.13)

3 Concluding remarks

From that expression of Morozov’s formula, it is rather easy to recover any individual correlator by taking residues:

D UijUji† E = Res x→xi Res y→xj tr 1 x− XU 1 y− Y U † (3.14) Notice also that 2.13 is very similar to what was found in [2] after integration over X and Y .

Notice that by expanding the determinant in 2.13 along its last column, one can find a recursion relation relating U(N) to U(N − 1) integrals, which is equivalent to Shatashvili’s approach [3].

Aknowledgements: the author wants to thank M. Bertola and J. Harnad for discus-sions about that topic. The author thinks that this short note should be seen as an addendum to the article [2].

References

[1] A. Morozov, “Pair correlator in the Itzykson–Zuber Integral”, Modern Phys. Lett. A 7, no. 37 3503–3507 (1992).

[2] M. Bertola, B. Eynard, ”Mixed Correlation Functions of the Two-Matrix Model”, SPHT T03/028, CRM-2961 (2003). J. Phys. A36 (2003) 7733-7750, xxx, hep-th/0303161.

[3] S. L. Shatashvili, “Correlation Functions in the Itzykson–Zuber Model”, Comm. Math. Phys. 154 421–432 (1993).

[4] C. Itzykson and J.B. Zuber, “The planar approximation (II)”, J. Math. Phys. 21, 411 (1980).