Consider the subspace Wn of L 2 (C, dA) consisting of all weighted polynomials W (z) = P (z)•e − 1 2 nQ(z) , where P (z) is a holomorphic polynomial of degree at most n−1, Q(z) = Q(z, z) is a fixed, real-valued function called the "external potential", and dA = − 1 2πi dz ∧ dz is normalized Lebesgue measure in the complex plane C.We study large n asymptotics for the reproducing kernel Kn(z, w) of Wn; 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 Ĉ \ S containing ∞, leaving aside such cases which are at this point well-understood.For the Ginibre kernel, corresponding to 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. With an extra twist we also obtain an off-diagonal estimate when both z and w are on the boundary ∂U :The bound is sharp for the Ginibre kernel, and is believed to be sharp in general.For given z ∈ U we also study the Berezin probability measures dµn,z(w) = |Kn(z,w)| 2 Kn(z,z) dA(w); we prove a quantitative approximation formula which shows that µn,z converges as n → ∞ to the harmonic measure of U evaluated at z, with "Gaussian convergence". In a sense this unifies earlier work on Berezin measures and root functions due to Ameur, Hedenmalm and Makarov, and to Hedenmalm and Wennman, respectively.In addition, we discuss the integrable structure underlying the whole theme, provided by the loop equation (Ward's identity), and we consider the problem of extending our results to disconnected droplets (archipelagos).

We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the "hard edge regime" where all disk boundaries are a distance of order 1 n away from the hard wall, and (b) the "semi-hard edge regime" where all disk boundaries are a distance of order 1 √ n away from the hard wall. As n → +∞, we prove that the moment generating function enjoys asymptotics of the form

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.