Zero distribution of complex orthogonal polynomials with respect to exponential weights (Q2016133)

The authors study the zeros of orthogonal polynomials with respect to a varying exponential weight \(\exp{(-nV(z)}\) with \(V(z)\) a polynomial. The weight is taken along an unbounded contour \(\Gamma\) in the complex plane, connecting two sectors in which \(\text{Re}\,V(z) \rightarrow +\infty\). The monic orthogonal polynomial \(P_n\) of degree \(n\) satisfies \[ \int_{\Gamma}\,z^kP_n(z)e^{-nV(z)}dz=0\text{ for }k=0,1,\ldots,n-1. \] Let now \(V\) be a polynomial of degree \(d\geq 2\), then there are exactly \(d\) sectors with each having an opening angle \(\pi /d\), such that \(\text{Re}\,V(z) \rightarrow +\infty\) in each of these sectors (counter clockwise denoted by \(S_1,S_2,\ldots,S_d\), with \(S_1\) satisfying \(0<\text{arg}\,z < \pi /d\)), interlacing with \(d\) sectors where \(\text{Re}\,V(z) \rightarrow -\infty\). The main results are then given in: { Theorem 1.1}. Let \(V(z)=-{iz^3\over 3}+iKz\) with \(K\in\mathbb{R}\) and let \({\mathcal T}={\mathcal T}_{2,1}\) be the family of infinite contours (i.e., closed contours on the Riemann sphere through the north pole) that connect the sectors \(S_2\) to \(S_1\). Then there is a unique critical value \(K^{\ast}\) such that (a) \(K<K^{\ast}\): the zeros of \(P_n\) accumulate on one analytic arc, (b) \(K=K^{\ast}\): the zeros of \(P_n\) accumulate on an arc which is not analytic at the point of intersection with the imaginary axis, (a) \(K>K^{\ast}\): the zeros of \(P_n\) accumulate on two disjoint arcs. The constant \(K^{\ast}\) is determined by \[ K^{\ast}=(v^{\ast})^{1/3}-(v^{\ast})^{-2/3}\approx 1.0005424, \] with \(v^{\ast}\approx 3.15037174\) the unique solution of \[ -3v\ln{(2v)}+6v\ln{(\sqrt{4+2v}+2)}+(2-2v)\sqrt{4+2v}=0,\;v>0. \] { Theorem 1.2}. Let \(V(z))=-{iz^3\over 5}\). Then for both choices \({\mathcal T}_{3,1}\) (connecting \(S_3\) and \(S_5\)) and \({\mathcal T}_{4,5}\) (connecting \(S_4\) and \(S_5\)), the zeros of the orthogonal polynomials accumulate on one analytic arc. The proofs use results on \(S\)-curves (cf. \textit{A. A. Gonchar} and \textit{E. A. Rakhmanov} [Math. USSR, Sb. 62, No. 2, 305--348 (1989); translation from Mat. Sb., Nov. Ser. 13 (176; Zbl 0663.30039)] and Teichmüller's lemma Theorem 14.1 in [\textit{K. Strebel}, Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Band 5. Berlin etc.: Springer-Verlag. (1984; Zbl 0547.30001)]).