CRM-3169 (2004) Saclay-T04/019
arXiv:nlin/0410043v1 [nlin.SI] 20 Oct 2004
Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions1 M. Bertola†‡2 , B. Eynard†⋆3 and J. Harnad†‡4 † Centre de recherches math´ematiques, Universit´e de Montr´eal C. P. 6128, succ. centre ville, Montr´eal, Qu´ebec, Canada H3C 3J7 ‡
Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montr´eal, Qu´ebec, Canada H4B 1R6 ⋆
Service de Physique Th´eorique, CEA/Saclay Orme des Merisiers F-91191 Gif-sur-Yvette Cedex, FRANCE
Abstract The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probablities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions.
Contents 1 Introduction
2
2 Generalized orthogonal polynomials and partition functions 2.1 Orthogonality measures and integration contours . . . . . . . . . . 2.1.1 Definition of the boundary-free contours . . . . . . . . . . . 2.1.2 Definition of the hard-edge contours . . . . . . . . . . . . . 2.2 Recursion relations, derivatives and deformations equations . . . . 2.2.1 Existence of orthogonal polynomials and relation to random 2.2.2 Wave vector equations . . . . . . . . . . . . . . . . . . . . . 2.2.3 Wave vector of the second kind . . . . . . . . . . . . . . . .
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1 Research supported in part by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR du Qu´ ebec and EC ITH Network HPRN-CT-1999-000161. 2
[email protected] 3
[email protected] 4
[email protected]
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3 Folding 3.1 n-windows and Christoffel-Darboux formula . . . . . . . . 3.2 Folded version of the deformation equations for changes in 3.3 Folding of the endpoint deformations . . . . . . . . . . . . 3.4 Folded version of the recursion relations and ~∂x relations
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14 14 15 16 17
4 Spectral curve and spectral invariants 4.1 Virasoro generators and the spectral curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Spectral residue formulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Isomonodromic Tau function 5.1 Isomonodromic deformations and residue formula . . . . . . . . . . . . . . . . . . . . . . . 5.2 Traceless gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
The partition function for Hermitian random matrix models with measures that are exponentials of a polynomial potential was shown in [4] to be equal, within a multiplicative factor independent of the deformation parameters, to the Jimbo-Miwa–Ueno isomonodromic tau function [11] for the rank 2 linear differential system satisfied by the corresponding set of orthogonal polynomials. The results of [4] were in fact more general, in that polynomials orthogonal with respect to complex measures supported along certain contours in the complex plane were considered. These may be viewed as corresponding to unitarily diagonalizable matrix models in which the spectrum is constrained to lie on these contours. The purpose of the present work is to extend these considerations to the more general setting of complex measures whose logarithmic derivatives are arbitrary rational functions, the associated semiclassical orthogonal polynomials and generalized matrix models. By also including contours with endpoints, the latter viewed as further deformation parameters, the gap probability densities are included as special cases of partition functions. To place the results in context, we first briefly recall the main points of [4], restricting to the more standard case of Hermitian matrices and real measures. Consider orthogonal polynomials πn (x) ∈ −1 L2 (R, e−~ V (x) dx) supported on the real line, with the measure defined by exponentiating a real polynomial potential d X tJ J V (x) = x . (1-1) J J=1
(Here we assume V (x) is of even degree and with positive leading coefficient, although these restrictions are unnecessary in the more general setting of [4].) The small parameter ~ is usually taken as O(N −1 ) when considering the limit N → ∞. Any two consecutive polynomials satisfy a first order system of ODE’s d πn−1 (x) πn−1 (x) = Dn (x) , (1-2) ~ πn (x) πn (x) dx where Dn (x) is a 2 × 2 matrix with polynomial coefficients of degree at most d − 1 = deg(V ′ (x)). The infinitesimal deformations corresponding to changes in the coefficients {tJ } result in a sequence of Frobenius compatible, overdetermined systems of PDE’s ∂ πn−1 (x) πn−1 (x) = Tn,J (x) J = 1, . . . , d, (1-3) ~ πn (x) πn (x) ∂tJ
2
where the matrices Tn,J (x) are polynomials in x of degree J which satisfy the compatibility conditions ∂ d ~ − Tn,J (x), ~ − Dn (x) = 0 . (1-4) ∂tJ dx It follows that the generalized monodromy data of the sequence of rational covariant derivative operators d − Dn (x) are invariant under these deformations, and independent of the integer n. ~ dx This is a particular case of the general problem of rational isomonodromic deformation systems [11]. An important rˆole is played in this theory by the isomonodromic tau function τnIM associated with any solution of an isomonodromic deformation system. This function on the space of deformation parameters is obtained by integrating a closed differential whose coefficients are given by residues involving the fundamental solutions of the system. The main results of [4] were the following. First, the coefficients of the associated spectral curve, given by the characteristic equation of the matrix Dn (x), can be obtained by applying certain first order differential operators with respect to the deformation parameters (Virasoro generators) to ln(Zn (V )), where the partition function Zn (V ) of the associated n × n matrix model is Z 1 (1-5) dM exp − TrV (M ) . Zn (V ) := ~ Hn Second, this partition function is equal to the isomonodromic tau function up to a multiplicative factor that does not depend on the deformation parameters τnIM = Zn (V ) Fn .
(1-6)
The present work generalizes these results to the case of measures whose logarithmic derivatives are arbitrary rational functions, including those supported on curve segments in which the endpoints may play the rˆole of further deformation parameters. The latter are of importance in the calculation of gap probabilities in matrix models [17] since these may, in this way, using measures supported on such segments be put on the same footing as partition functions [6]. A Frobenius compatible system of first order differential and deformation equations satisfied by the corresponding orthogonal polynomials is derived (Propositions 3.2, 3.4) and the coefficients of the associated spectral curve are again shown to be obtained by applying suitable Virasoro generators to ln(ZN (V )) (Theorems 4.1–4.2). A formula that generalizes (1-6) is also derived (Theorem 5.1 ): τnIM = Zn (V )Fn (V ) ,
(1-7)
where the factor Fn (V ) is an explicitly computed function of the deformation parameters determining V , which can in fact be eliminated by a making a suitable scalar gauge transformation. The results of Theorems 4.1, 4.2 give a precise meaning, for finite n, to formulæ that are usually derived in the asymptotic limit n → ∞ (~n ∼ O(1)) through saddle point computations, relating the free energy to the asymptotic spectral curve. It is well known [9] that the free energy in the large n limit is given by solving a minimization problem (in the Hermitian matrix model) Z Z Z F0 := − lim ~2 ln Zn = min V (x)ρ(x)dx − ρ(x)ρ(x′ ) ln |x − x′ | , (1-8) n→∞
ρ(x)≥0
giving the equilibrium density ρeq for the eigenvalue distribution. If, for instance, the potential is a real polynomial bounded from below, it is known [8] that the support of the equilibrium density is a union of finite segments I ⊂ R. The density ρeq is obtained from the variational equation Z ρeq (x)dx = V ′ (x) (1-9) 2P x − x′ 3
(within I), and is related to the resolvent by Z ρeq (x) 1 = dx , z ∈C\I . ω(z) := ~ lim Tr n→∞ M −z x−z I
(1-10)
In terms of this, the spectral density may be recovered as the jump-discontinuity of ω(z) across I, and all its moments are given by Z
xJ 1 zJ J ∂tJ F0 = lim ~ TrM = dx ρeq (x) = − res ω(z)dz . (1-11) z=∞ J J n→∞ J I The function y = −ω(x) satisfies an algebraic relation given by y 2 = yV ′ (x) + R(x) ,
(1-12)
where R(x) is a polynomial of degree less than V ′ (x) that is uniquely determined by the consistency of (1-9) and (1-12). The point to be stressed here is that this asymptotic spectral curve should be compared with the spectral curve of Theorem 4.1, given by the characteristic equation (4-29), which also contains all the relevant information about the finite n case. In the n → ∞ limit, logarithmic derivatives of the partition function are expressed in (1-11) as residues of the meromorphic differentials z k ydz on the curve. The same formulae are shown in Theorem 4.2 to hold as exact relations for the finite n case if we replace the “asymptotic” spectral curve by the spectral curve given by the characteristic equation of the matrix Dn (x). The paper is organized as follows. In section 2 the problem is defined in terms of polynomials orthogonal with respect to an arbitrary semiclassical measure supported on complex contours, and the corresponding generalized matrix model partition functions. In section 2.2 the recursion relations, differential systems and deformation equations which these satisfy are expressed in terms of the semi-infinite “wave vector” formed from the orthogonal polynomials. In section 3 the notion of “folding” is introduced and used (Propositions 3.2–3.4) to express the preceding equations as an infinite sequence of compatible overdetermined 2 × 2 systems of linear differential equations and recursion relations satisfied by pairs of consecutive orthogonal polynomials. In section 4 the results of folding are used to express the spectral curve in terms of logarithmic derivatives of the partition function and it is shown that the n → ∞ relation between the free energy and the spectral curve is also valid as an exact result for finite n. In section 5 the definition of the isomonodromic tau function [11] is recalled and it is computed by relating it to the spectral invariants of the rational matrix generalizing Dn (x) in (1-2). These invariants are shown to give the logarithmic derivatives of the tau functions in terms of residues of meromorphic differentials on the spectral curves through formaulae that are nearly identical to those for the partition function, This leads to the main result, Theorem 5.1, which gives the explicit relation between Zn and τnIM .
2 2.1
Generalized orthogonal polynomials and partition functions Orthogonality measures and integration contours
Given a measure on the real line, the associated orthogonal polynomials are those that diagonalize the quadratic form associated to the corresponding (complex) moment functional; i.e., the linear form obtained by integration with respect to the measure. L : C[x] p(x)
→ C. R 7→ L(p(x)) = R p(x)dµ(x) . 4
(2-1)
A natural generalization consists of including moment functionals that are expressed by integration along more general contours in the complex x plan, with respect to a complex measure defined by locally analytic weight functions that may have isolated essential singular points and complex power-like branch points. We thus consider linear forms on polynomials given by integrals of the form Z L(p(x)) = p(x)µ(x)dx κ
1
µ(x) = e− ~ V (x) K X V (x) := Tr (x) ,
(2-2)
r=0
where T0 (x) := t0,0 +
d0 X t0,J J x J
J=1
Tr (x) :=
dr X
J=1
tr,J − tr,0 ln(x − cr ) J(x − cr )J ′
− ~∂x ln µ(x) = V (x) =
dX 0 −1
J−1
t0,J x
J=1
−
K dX r +1 X r=1 J=1
tr,J−1 , (x − cr )J
(2-3)
R and the symbol κ denotes integration over linear combinations of contours on which the integrals are convergent, as explained below. This class of linear functionals is sometimes referred to as semiclassical moment functionals [3, 14, 15]. We consider the corresponding monic generalized orthogonal polynomials pn (x), which satisfy Z pn (x)pm (x)µ(x)dx = hn δnm .
(2-4)
κ
If all the contours are contained in the real axis and the weight is real and positive, we reduce to the usual notion of semiclassical orthogonal polynomials. The small parameter ~ introduced in (2-2) is not of essential importance here; it is only retained in the formulæ below to recall that, when taking the large n limit, it plays the rˆole of small parameter for which ~n remains finite as n → ∞. (j) To describe the contours of integration, we first define sectors Sr , r = 0, . . . K, k = 1, . . . dr . around the points cr for which dr > 1 (c0 := ∞) in such a way that +∞ . ℜ (V (x)) x −→ →c , x∈
r Sr(j)
(2-5)
The number of sectors for each pole in V is equal to the degree of that pole; that is, d0 for the pole at infinity and dr for the pole at cr . Explicitly 2kπ − arg(t0,d0 ) + π2 2kπ − arg(t0,d0 ) − π2 (0) , (2-6) < arg(x) < Sk := x ∈ C; d0 d0 k = 0. . . d0 − 1 ; 2kπ + arg(tr,dr ) − π2 2kπ + arg(tr,dr ) + π2 (r) Sk := x ∈ C; , (2-7) < arg(x − cr ) < dr dr k = 0, . . . , dr − 1, r = 1, . . . , K . These sectors are defined in such a way that approaching any of the essential singularities of µ(x) (i.e. a cr such that dr > 0) within them, the function µ(x) tends to zero faster than any power.
5
2.1.1
Definition of the boundary-free contours
The definition of the contours follows [16] (see fig. 1). 1. For any cr for which there is no essential singularity in the measure (i.e. dr = 0), there are two subcases: (a) For the cr ’s that are branch points or poles in µ (i.e., tr,0 ∈ / N), we take a loop starting at (0) infinity in some fixed sector Sk encircling the singularity and going back to infinity in the same sector. (Note that if cr is just a pole; i.e., tr,0 ∈ −N+ , the contour could equivalently be taken as a circle around cr .) (b) For the cr ’s that are regular points (tr,0 ∈ N ), we take a line joining cr to infinity, approaching (0) ∞ in a sector Sk as before. 2. For any cr for which there is an essential singularity in µ (i.e. , dr > 0) we define dr contours (r) (r) starting from cr in the sector Sk and returning to it in the next sector Sk+1 . Also, if tr,0 ∈ / Z, we (0)
join the singularity cr to ∞ by a path approaching ∞ within one fixed sector Sk . (0)
3. For c0 := ∞, we take d0 − 1 contours starting at c0 in the sector Sk (0) next sector Sk+1 .
and returning at c0 in the
Note that, with these definitions, the integrals involved are convergent and we can perform integration by parts. Moreover, any contour in the complex plane for which the integral of µ(x)p(x)dx is convergent for all polynomials p(x) is equivalent to a linear combination of the contours defined above, no two of which are, in this sense, equivalent. 2.1.2
Definition of the hard-edge contours
We also include some additional contpours in the complex plane {m)j}j=1,...,L , starting at some points (0) aj , j = 1 . . . L and going to ∞ within one of the sectors Sk . These could be viewed as corresponding to additional points in 1(b) for which both dr = 0 and tr,0 = 0, but we prefer to deal with them separately since integration by parts on these PKcontours does give a contribution. In total there are S := d0 + r=1 (dr + 1) boundary-free contours σℓ , , ℓ = 1, . . . , S and L hard-edge contours mh , h = 1, . . . , L. The moment functional is an arbitrary linear combination of integrals taken along these contours Z Z Z S L X X . (2-8) κL+j + κj := κ
j=1
mj
j=1
σj
Note that, by taking appropriate linear combinations of the contours, we could alternatively have had contours consisting of finite segments joining the points aj .
6
Sk
L
*c2
a3
a2
c1
a1
c3
Figure 1: The types of contours considered in the x Riemann sphere P1 . Here we have c1 with d1 = 3 and c2 with d2 = 0, t2,0 ∈ / Z (logarithmic singularity in the potential), c3 with d3 = 0, t3,0 ∈ N and the degree of the potential at infinity c0 = ∞ is d0 = 5. The essential singularity in µ at c1 is of the form exp (x − c1 )−3 and there is also a cut extending from c1 to ∞ if t1,0 ∈ / Z. The point c2 is a branch point of µ(x) since t2,0 ∈ / Z, and the cut extends to infinity “inside” the contour (as shown here). If it were a pole (t2,0 ∈ −N+ ), the contour would be replaced by a circle around it. The point c3 is a regular point with t3,0 ∈ N× , and the contour extending from it to infinity is no different from the ones starting at the regular points a1 , a2 , a3 . The latter are the “hard-edge” (0) segments joining the points a1 , a2 and a3 to ∞ within one of the sectors Sk .
2.2 2.2.1
Recursion relations, derivatives and deformations equations Existence of orthogonal polynomials and relation to random matrices
Recall [7] that orthogonal polynomials satisfying (2-4) exist provided all the Hankel determinants formed from the moments are nonzero: Z i+j ∆n (κ) := det x µ(x)dx 6= 0 , ∀n ∈ N. (2-9) κ
0≤i,j≤n−1
Since the ∆n (κ)’s are homogeneous polynomials in the coefficients κj , the zero locus excluded by (2-9) is of zero measure (in the space of κj ’s), and hence “generically” the conditions (2-9) are fulfilled. The development to follow will in fact only involve orthogonal polynomials up to some arbitrarily large fixed degree, say N , and hence the conditions ∆n (κ) = 0 , n ≤ N − 1 determine a Zariski closed set in {κj }, (and a closed set of measure zero in the space of coefficients of V ). The orthogonal polynomials considered here are related to models of unitarily diagonalizable random matrices M ∈ gl(n, C) with spectra supported on the contours defined above. More specifically we have the partition function Z 1 Zn := Cn dM e− ~ TrV (M) spec(M)∈κ
7
=
Z
κ
dx1 · · ·
Z
1
dxn ∆(x)2 e− ~ κ
= n!∆n (κ, V ) = n!
n−1 Y
Pn
j=1
V (xj )
hj ,
(2-10)
(2-11)
j=0
where CN := R
1
U(n) dU
is the inverse of the U (n) group volume, and
∆(x) :=
Y (xi − xj )
(2-12)
i
