Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials

Abstract

Consider the subspace $${{{\mathscr {W}}}_{n}}$$ W n of $$L^2({{\mathbb {C}}},dA)$$ L 2 ( C , d A ) consisting of all weighted polynomials $$W(z)=P(z)\cdot e^{-\frac{1}{2}nQ(z)},$$ W ( z ) = P ( z ) · e - 1 2 n Q ( z ) , where P(z) is a holomorphic polynomial of degree at most $$n-1$$ n - 1 , $$Q(z)=Q(z,{\bar{z}})$$ Q ( z ) = Q ( z , z ¯ ) is a fixed, real-valued function called the “external potential”, and $$dA=\tfrac{1}{2\pi i}\, d{\bar{z}}\wedge dz$$ d A = 1 2 π i d z ¯ ∧ d z is normalized Lebesgue measure in the complex plane $${{\mathbb {C}}}$$ C . We study large n asymptotics for the reproducing kernel $$K_n(z,w)$$ K n ( z , w ) of $${{\mathscr {W}}}_n$$ W n ; this depends crucially on the position of the points z and w relative to the droplet S, i.e., the support of Frostman’s equilibrium measure in external potential Q. We mainly focus on the case when both z and w are in or near the component U of $$\hat{{{\mathbb {C}}}}\setminus S$$ C ^ \ S containing $$\infty $$ ∞ , leaving aside such cases which are at this point well-understood. For the Ginibre kernel, corresponding to $$Q=|z|^2$$ Q = | z | 2 , we find an asymptotic formula after examination of classical work due to G. Szegő. Properly interpreted, the formula turns out to generalize to a large class of potentials Q(z); this is what we call “Szegő type asymptotics”. Our derivation in the general case uses the theory of approximate full-plane orthogonal polynomials instigated by Hedenmalm and Wennman, but with nontrivial additions, notably a technique involving “tail-kernel approximation” and summing by parts. In the off-diagonal case $$z\ne w$$ z ≠ w when both z and w are on the boundary $${\partial }U$$ ∂ U , we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic $$\begin{aligned} K_n(z,w)\sim \sqrt{2\pi n}\,\Delta Q(z)^{\frac{1}{4}}\,\Delta Q(w)^{\frac{1}{4}}\,S(z,w) \end{aligned}$$ K n ( z , w ) ∼ 2 π n Δ Q ( z ) 1 4 Δ Q ( w ) 1 4 S ( z , w ) where S(z, w) is the Szegő kernel, i.e., the reproducing kernel for the Hardy space $$H^2_0(U)$$ H 0 2 ( U ) of analytic functions on U vanishing at infinity, equipped with the norm of $$L^2({\partial }U,|dz|)$$ L 2 ( ∂ U , | d z | ) . Among other things, this gives a rigorous description of the slow decay of correlations at the boundary, which was predicted by Forrester and Jancovici in 1996, in the context of elliptic Ginibre ensembles.

Large gap asymptotics on annuli in the random normal matrix model

“…The random normal matrix model has been extensively studied over the years, see e.g. [20,32] for early works, [7,38,53,66] for smooth linear statistics, [9,17,30,36,65,74] for non-smooth linear statistics, and [6,11,45,54,55] for recent investigations on planar orthogonal polynomials. Despite such progress, the problem of determining large gap asymptotics in this model has remained an outstanding problem.…”

Exponential moments for disk counting statistics at the hard edge of random normal matrices

“…// is a polynomial of order 2j C 1 in . The following lemma follows from a result of Tricomi [75], see also [7]. However, in contrast to [7,75], the coefficients appearing in Lemma A.4 below are written in a non-recursive way.…”

“…The following lemma follows from a result of Tricomi [75], see also [7]. However, in contrast to [7,75], the coefficients appearing in Lemma A.4 below are written in a non-recursive way. Here we give a short proof relying on Lemmas A.2 and A.3.…”