The Szegő curve and Laguerre polynomials with large negative parameters

Abstract

We study the asymptotic zero distribution of the rescaled Laguerre polynomials, Lwith the parameter α n varying in such a way that lim n→∞ α n /n = −1. The connection with the so-called Szegő curve is shown.

Fine structure in the large n limit of the non-Hermitian Penner matrix model

“…However, the case t = 1 can be independently studied in terms of the asymptotic zero distribution of Laguerre polynomials in the limit (48) with A = −1. This zero distribution was determined in [28] and also exhibits a fine structure (see Theorem 1 of [28]). Indeed, the support of the zero density ρ(z) depends on the sequence α n , but now through the parameter m = lim…”

“…Riemann-Hilbert and steepest-descent methods [15,26,27,28] have permitted the complete characterization of the asymptotic zero distribution ρ L (z) of the scaled Laguerre polynomials L …”

Equilibrium measures in the presence of certain rational external fields

“…, q} ⊂ N . Indeed, for t = 0, the left-hand member (7) reduces to ϕ and, thus, the support of the equilibrium measure starts at one or several of the critical points of the external field; in a similar fashion, these critical points of ϕ are the initial conditions of the dynamical system (11). Regarding the second question, we have, Theorem 2.2.…”

Root-counting Measures of Jacobi Polynomials and Topological Types and Critical Geodesics of Related Quadratic Differentials

“…For a point x 1 x 1 a complex chart can be assigned as follows: 14) where the branches of the radicals are taken such that ζ(w) > 0 when w is real such that w > 1 or w < 0. Similarly, to assign a complex chart to a point x 2 + ih 1 x 2 + ih, we use the following mapping: 15) with appropriate branches of the radicals. To a point of R corresponding to an infinite boundary point b 1 , a complex chart can be assigned via the function ζ = exp(−2πiw) for w such that 0 ≤ w ≤ 1, w < 0, (8.16) which maps the half-strip {w : 0 ≤ w ≤ 1, w < 0} onto the unit disc punctured at ζ = 0.…”

“…Two main themes of this work are asymptotic behavior of zeros of certain polynomials and topological properties of related quadratic differentials. The study of asymptotic root distributions of hypergeometric, Jacobi, and Laguerre polynomials with variable real parameters, which grow linearly with degree, became a rather hot topic in recent publications, which attracted attention of many authors [14], [15], [16], [17], [18], [22], [24], [25], [27]. In this paper, we survey some known results in this area and present some new results keeping focus on Jacobi polynomials.…”