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.
In this note we study a minimization problem for a vector of measures subject to a prescribed interaction matrix in the presence of external potentials. The conductors are allowed to have zero distance from each other but the external potentials satisfy a growth condition near the common points.We then specialize the setting to a specific problem on the real line which arises in the study of certain biorthogonal polynomials (studied elsewhere) and we prove that the equilibrium measures solve a pseudoalgebraic curve under the assumption that the potentials are real analytic. In particular, the supports of the equilibrium measures are shown to consist of a finite union of compact intervals.
Based on the work of Itzykson and Zuber on Kontsevich's integrals, we give a geometric interpretation and a simple proof of Zhou's explicit formula for the Witten-Kontsevich tau function. More precisely, we show that the numbers A A by-product of our study indicates an interesting relation between the matrix-valued affine coordinates for the Witten-Kontsevich tau function and the V -matrices associated to the R-matrix of Witten's 3-spin structures.
Abstract. Generalized Vorob'ev-Yablonski polynomials have been introduced by Clarkson and Mansfield in their study of rational solutions of the second Painlevé hierarchy. We present new Hankel determinant identities for the squares of these special polynomials in terms of Schur polynomials. As an application of the identities, we analyze the roots of generalized Vorob'ev-Yablonski polynomials and provide formulae for the boundary curves of the highly regular patterns observed numerically in [8].
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