We obtain the strong asymptotics of polynomials p n (λ), λ ∈ C, orthogonal with respect to measures in the complex plane of the formwhere s is a positive integer, t is a complex parameter, and dA stands for the area measure in the plane. This problem has its origin in normal matrix models. We study the asymptotic behavior of p n (λ) in the limit n, N → ∞ in such a way that n/N → T constant. Such asymptotic behavior has two distinguished regimes according to the topology of the limiting support of the eigenvalues distribution of the normal matrix model. If 0 < |t| 2 < T /s, the eigenvalue distribution support is a simply connected compact set of the complex plane, while for |t| 2 > T /s, the eigenvalue distribution support consists of s connected components. Correspondingly, the support of the limiting zero distribution of the orthogonal polynomials consists of a closed contour contained in each connected component. Our asymptotic analysis is obtained by reducing the planar orthogonality conditions of the polynomials to equivalent contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann-Hilbert problem by the Deift-Zhou nonlinear steepest descent method.
In this note the logarithmic energy problem with external potential |z| 2n + tz d +tz d is considered in the complex plane, where n and d are positive integers satisfying d ≤ 2n. Exploiting the discrete rotational invariance of the potential, a simple symmetry reduction procedure is used to calculate the equilibrium measure for all admissible values of n, d and t.It is shown that, for fixed n and d, there is a critical value |t| = tcr such that the support of the equilibrium measure is simply connected for |t| < tcr and has d connected components for |t| > tcr.
After an overview of variational principles for discrete gravity, and on the basis of the approach to conformal transformations in a simplicial PL setting proposed by Luo and Glickenstein, we present at a heuristic level an improved scheme for addressing the gravitational (Euclidean) path integral and geometrodynamics.
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