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Dedicated to Nico Temme on the occasion of his 65th birthday. AbstractWe give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. The leading role is played by the differential equation satisfied by these polynomials. Some new developments, applications and open problems are presented.

In this paper we study the asymptotics (as n → ∞) of the sequences of Laguerre polynomials with varying complex parameters α depending on the degree n. More precisely, we assume that α n = nA n , and lim n A n = A ∈ C. This study has been carried out previously only for α n ∈ R, but complex values of A introduce an asymmetry that makes the problem more difficult.The main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality, which requires the analysis of the global structure of trajectories of an associated quadratic differential on the complex plane, which may have an independent interest.While the weak asymptotics is obtained by reduction to the theorem of GoncharRakhmanov-Stahl, the strong asymptotic results are derived via the non-commutative steepest descent analysis based on the Riemann-Hilbert characterization of the Laguerre polynomials.

Motivated by the study of the asymptotic behavior of Jacobi polynomials P (nA,nB) n n with A ∈ C and B > 0 we establish the global structure of trajectories of the related rational quadratic differential on C. As a consequence, the asymptotic zero distribution (limit of the root-counting measures of P (nA,nB) n n ) is described. The support of this measure is formed by an open arc in the complex plan (critical trajectory of the aforementioned quadratic differential) that can be characterized by the symmetry property of its equilibrium measure in a certain external field.