Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

Abstract

Let Λ R denote the linear space over R spanned by z k , k ∈ Z. Define the real inner product (with varying exponential weights). . } with respect to · · ·, · · · L yields the even degree and odd degree orthonormal Laurent polynomialsn > 0, and φ 2n+1 (z) = ξ n , φ 2n (z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequenceare obtained by formulating the even degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on R, and then extracting the largen behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2,3].

Matrix biorthogonal polynomials: Eigenvalue problems and non-Abelian discrete Painlevé equations

“…Later, Deift and Zhou combined these ideas with a non-linear steepest descent analysis in a series of papers [25,26,28,29] which was the seed for a large activity in the field. To mention just a few relevant results let us cite the study of strong asymptotic with applications in random matrix theory, [25,27], the analysis of determinantal point processes [22,23,45,46], orthogonal Laurent polynomials [49,50] and Painlevé equations [24,41].…”

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

“…Deift and Zhou combined these ideas with a nonlinear steepest descent analysis in a series of important works [6][7][8][9] that, as a by-product, generated a large activity in the field. To mention just a few relevant results, let us cite the study of strong asymptotic with applications in random matrix theory, 6,10 the analysis of determinantal point processes, [11][12][13][14] orthogonal Laurent polynomials, 15,16 and Painlevé equations. 17,18 For the case of OPUC, an RHp was discussed in Ref.…”

Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlevé IV

“…Later, Deift and Zhou combined these ideas with a non-linear steepest descent analysis in a series of papers [27,28,30,31] which was the seed for a large activity in the field. To mention just a few relevant results let us cite the study of strong asymptotic with applications in random matrix theory [27,29], the analysis of determinantal point processes [24,25,48,49], orthogonal Laurent polynomials [51,52] and Painlevé equations [26,45].…”