Complex Path Integral Representation for Semiclassical Linear Functionals

2009

Commun. Math. Phys.

132

The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange's divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno provided an algebraic construction of its derivatives with respect to isomonodromic times. However the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes' parameters) was not derived. We fill the gap by providing a (simpler and more general) description in which all the parameters of the problem (monodromy-changing and monodromy-preserving) are dealt with at the same level. We thus provide variational formulae for the isomonodromic tau function with respect to the (generalized) monodromy data. The construction applies more generally: given any (sufficiently well-behaved) family of Riemann-Hilbert problems (RHP) where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form Ω (not necessarily closed) on the deformation space (Malgrange's differential), defined off Malgrange's divisor. We then introduce the notion of discrete Schlesinger transformation: it means that we allow the solution of the RHP to have poles (or zeros) at prescribed point(s). Even if Ω is not closed, its difference evaluated along the original solution and the transformed one, is shown to be the logarithmic differential (on the deformation space) of a function. As a function of the position of the points of the Schlesinger transformation, yields a natural generalization of Sato formula for the Baker-Akhiezer vector even in the absence of a tau function, and it realizes the solution of the RHP as such BA vector.Some exemplifications in the setting of the Painlevé II equation and finite Töplitz/Hankel determinants are provided.

Section: Hankel and Shifted Töplitz Determinantsmentioning

confidence: 99%

The Dependence on the Monodromy Data of the Isomonodromic Tau Function

2009

Commun. Math. Phys.

132

“…In particular, as in the standard case, the orthogonal polynomials may be shown to be equal to the expectation values of the characteristic polynomials in such models 14) and all correlation functions between the eigenvalues may be expressed as determinants in terms of the standard Christoffel-Darboux kernel formed from them…”

Section: Existence Of Orthogonal Polynomials and Relation To Random Mmentioning

confidence: 99%

“…This class of linear functionals is sometimes referred to as semiclassical moment functionals [3,14,15]. We consider the corresponding monic generalized orthogonal polynomials p n (x), which satisfy…”

Section: Orthogonality Measures and Integration Contoursmentioning

confidence: 99%

“…The deformation equations (2-72), (2-73) (2-77) are equivalent to the infinite sequence of 2 × 2 equations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) where the folded matrix of the deformation is defined by…”

Section: Proposition 32mentioning

confidence: 99%

“…Remark 3.1 Note that formula (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) for the deformation of the measure in Proposition 3.2, as well as those below, (3-36), (3-37), which are obtained through folding of the ∂x operator, could also be derived for arbitrary locally analytic potentials V (x), provided all the integrals involved are convergent [13]. However applicability of the subsequent isomonodromic analysis would be lost if the derivatives were not rational, since the resulting deformation equations would then have essential singularities.…”

Section: Proposition 32mentioning

confidence: 99%

See 2 more Smart Citations

Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

Eynard

Harnad

2006

Commun. Math. Phys.

122

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.

Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane

Balogh

Lee

et al. 2014

Comm Pure Appl Math

We consider the orthogonal polynomials fP n .´/g with respect to the measure j´ aj 2Nc e N j´j 2 dA.´/ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with N . The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the g-function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region K of the complex plane. The third measure is the harmonic measure of K c with a pole at 1. This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the n th orthogonal polynomial times the orthogonality measure, i.e., jP n .´/j 2 j´ aj 2Nc e N j´j 2 dA.´/.The compact region K that is the support of the second measure undergoes a topological transition under the variation of the parameter t D n=N ; in a double scaling limit near the critical point given by t c D a.a C 2 p c/, we observe the Hastings-McLeod solution to Painlevé II in the asymptotics of the orthogonal polynomials.