Moment determinants as isomonodromic tau functions

Moment determinants as isomono dromic tau functions M. Bertola ‡ ,] 1 2 ‡ Dep artment of Mathematics and Statistics, Conc or dia University 1455 de Maisonneuve W., Montr ´ eal, Qu ´ eb e c, Canada H3G 1M8 ] Centr e de r e cher ches math ´ ematiques, Universit ´ e de Montr ´ eal Abstract W e consider a wide class of determinants whose en tries are moments of the so-called semiclassical func- tionals and w e sho w that they are tau functions for an appropriate isomono dromic family which dep ends on the parameters of the sym b ols for the functionals. This shows that the v anishing of the tau-function for those systems is the obstruction to the solv ability of a Riemann–Hilbert problem asso ciated to certain classes of (m ultiple) orthogonal p olynomials. The determinan ts include H¨ ank el, T¨ oplitz and shifted- T¨ oplitz determinants as well as determinants of bimoment functionals and the determinants arising in the study of multiple orthogonality . Some of these determinants app ear also as partition functions of random matrix mo dels, including an instance of a t wo-matrix mo del. Keyw ords: (Multiple) Orthogonal p olynomials, Random matrix theory , Sc hlesinger transforma- tions, Riemann-Hilb ert problems, Isomonodromic deformations AMS-MSC2000: 05E35, 15A52 Con ten ts 1 In tro duction 2 1.1 Organization of the pap er and outline of the pro of . . . . . . . . . . . . . . . . . . . . . . 4 2 Isomono dromic deformations: a (short) review 5 2.1 “T rivial” isomono dromic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Sc hlesinger transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Mark ov functions for semiclassical moment functionals 9 4 H¨ ank el determinants: orthogonal p olynomials 9 4.1 Riemann-Hilb ert problem of orthogonal p olynomial . . . . . . . . . . . . . . . . . . . . . . 9 4.2 A problem with constant jumps: ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Shifted T oeplitz determinants: biorthogonal Laurent and Szeg¨ o p olynomials. 12 5.1 Constan t-jump RHP and ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1 W ork supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). 2 b ertola@crm.umon treal.ca 1 6 Bimomen t determinants: Cauch y biorthogonal p olynomials 15 6.1 Constan t-jump RHP and ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7 Multi-measure momen t matrices: multiple orthogonal p olynomials 18 7.1 Constan t jump RHP and ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.1.1 Semiclassical weigh ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 8 Conclusion 21 1 In tro duction The connection b et ween orthogonal p olynomials and mono drom y preserving deformations of diﬀeren tial equations has b een observ ed rep eatedly [ 25 , 18 , 19 , 16 , 20 , 5 , 6 ]. It can b e used to pro duce special solution to P ainlev ´ e equations by c ho osing appropriately the measure of orthogonality and its dep endence on deformation parameters [ 25 ]. In the eighties the notion of isomono dromic tau function was developed [ 23 , 21 , 22 ] for systems of mono drom y preserving deformations of rational connections (under some genericit y assumptions); the philosophical attitude was that –in the con text of Birkhoﬀ–Riemann–Hilb ert problems– the tau-function should play the same rˆ ole as the classical Theta function pla ys for the Jacobi in version problem on algebraic curves. Indeed the v anishing of a Theta-function on the Ab el map of a divisor determines its “spec ialt y”, namely the (im)p ossibilit y of ﬁnding suitable meromorphic functions with prescrib ed divisor. Similarly the v anishing of the isomono dromic tau function indicates that a Riemann–Hilb ert problem is not solv able. This was prov ed ﬁrst for the Sc hlesinger isomono dromic equations by Malgrange [ 26 , 27 , 28 ] and more recen tly for connections with irregular singularities by P almer [ 33 ] (although under some genericity assumptions). In this p erspective a connection b et ween H¨ ank el determinants and isomono dromic tau functions is not surprising since the v anishing of the H¨ ankel determinant of the moments of a measure determines whether the orthogonal p olynomials exist or not and -on the other hand- their existence is equiv alen t to the solv abilit y of a RHP [ 19 ]. The precise functional relationship b et ween these (and similar) determinants and isomono dromic tau functions was established in [ 4 , 5 ] for the H¨ ank el case and in [ 20 , 6 ] for shifted T o eplitz determinan ts of so-called semiclassic al moment functionals [ 25 , 29 , 30 ]. In a seemingly distant area, moment determinants app ear as partition functions for random matrix mo dels [ 31 , 32 ], which are known to b e tau functions (in a diﬀerent sense) for the KP , T o da or 2T o da hierarc hies, dep ending on the case. F or example the partition function of the Hermitean matrix mo del deﬁned as Z 1 M M n := Z H n d M e − T r V ( M ) (1.1) where M is a Hermitean matrix and d M is the Leb esgue measure on the vector space H n of Hermitean matrices of size n × n ; here V ( x ) represents any scalar function, but for the sake of the example we can tak e it to b e a p olynomial. 2 By means of Dyson-Mehta theorem [ 32 ] it is known that Z 1 M M n is (up to an inessen tial normalization constan t) the same as a H¨ ankel determinant for the momen ts of the measure (or weigh t) e − V ( x ) d x Z 1 M M n ∝ ∆ n := det[ µ i + j ] 0 ≤ i,j ≤ n − 1 , µ j := Z x j e − V ( x ) d x (1.2) In addition the same quan tity can be expressed as a multiple integral ov er the sp ectra Z 1 M M n ∝ Z d x 1 . . . Z d x n ∆ 2 ( X )e − P n j =1 V ( x j ) , ∆( X ) := Y j

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