Grupo de Análisis Matemático Aplicado (GAMA)http://hdl.handle.net/10016/58572016-07-02T05:38:26Z2016-07-02T05:38:26ZPainlevé I double scaling limit in the cubic random matrix modelDeaño Cabrera, AlfredoBleher, Pavelhttp://hdl.handle.net/10016/232682016-07-01T10:46:54Z2016-04-01T00:00:00ZPainlevé I double scaling limit in the cubic random matrix model
Deaño Cabrera, Alfredo; Bleher, Pavel
We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the N×NN×N Hermitian random matrix model with cubic potential. We prove that the recurrence coefficients admit an asymptotic expansion in powers of N−2/5N−2/5, and in the leading order the asymptotic behavior of the recurrence coefficients is given by a Boutroux tronquée solution to the Painlevé I equation. We also obtain the double scaling limit of the partition function, and we prove that the poles of the tronquée solution are limits of zeros of the partition function. The tools used include the Riemann&-Hilbert approach and the Deift&-Zhou nonlinear steepest descent method for the corresponding family of complex orthogonal polynomials and their recurrence coefficients, together with the Toda equation in the parameter space.
2016-04-01T00:00:00ZVarying discrete Laguerre-Sobolev orthogonal polynomials: Asymptotic behavior and zerosMañas Mañas, Juan FranciscoMarcellán, FranciscoMoreno Balcazar, Juan J.http://hdl.handle.net/10016/232672016-07-01T10:39:35Z2013-10-01T00:00:00ZVarying discrete Laguerre-Sobolev orthogonal polynomials: Asymptotic behavior and zeros
Mañas Mañas, Juan Francisco; Marcellán, Francisco; Moreno Balcazar, Juan J.
We consider a varying discrete Sobolev inner product involving the Laguerre weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and of their zeros. We are interested in Mehler-Heine type formulas because they describe the asymptotic differences between these Sobolev orthogonal polynomials and the classical Laguerre polynomials. Moreover, they give us an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other special functions. We generalize some results appeared very recently in the literature for both the varying and non-varying cases.
2013-10-01T00:00:00ZBases of the space of solutions of some fourth-order linear difference equations: applications in rational approximationMarcellán, FranciscoMendes, AnaPijeira Cabrera, Héctor Estebanhttp://hdl.handle.net/10016/232652016-07-01T10:42:32Z2013-10-01T00:00:00ZBases of the space of solutions of some fourth-order linear difference equations: applications in rational approximation
Marcellán, Francisco; Mendes, Ana; Pijeira Cabrera, Héctor Esteban
It is very well known that a sequence of polynomials {Q(n)(x)}(n=0)(infinity) orthogonal with respect to a Sobolev discrete inner product (s) = integral(I)fg d mu + lambda f(-1)(0)g'(0); lambda is an element of R+; where mu is a finite Borel measure and I is an interval of the real line, satisfies a five- term recurrence relation. In this contribution we study other three families of polynomials which are linearly independent solutions of such a five- term linear difference equation and, as a consequence, we obtain a polynomial basis of such a linear space. They constitute the analogue of the associated polynomials in the standard case. Their explicit expression in terms of {Q(n)(x)}(n=0)(infinity) using an integral representation is given. Finally, an application of these polynomials in rational approximation is shown.
2013-10-01T00:00:00ZStrong asymptotics for the Pollaczek multiple orthogonal polynomialsAptekarev, AlexandreLópez Lagomasino, GuillermoMartinez Finkelshtein, Andreihttp://hdl.handle.net/10016/232582016-06-30T09:48:56Z2015-11-01T00:00:00ZStrong asymptotics for the Pollaczek multiple orthogonal polynomials
Aptekarev, Alexandre; López Lagomasino, Guillermo; Martinez Finkelshtein, Andrei
The asymptotic properties of multiple orthogonal polynomials with respect to two Pollaczek weights with different parameters are considered. This set of weights is a Nikishin system generated by two measures with unbounded supports; moreover, the second measure is discrete. During the last years, multiple orthogonal polynomials with respect to Nikishin systems of this type have found wide applications in the theory of random matrices. Strong asymptotic formulas for the polynomials under consideration are obtained by means of the matrix Riemann-Hilbert method.
2015-11-01T00:00:00Z