Bulk asymptotics for polyanalytic correlation kernels

Abstract

For a weight function Q:C→R and a positive scaling parameter m, we study reproducing kernels Kq,mQ,n of the polynomial spacesAq,mQ,n2:=spanC{z¯rzj|0⩽r⩽q−1,0⩽j⩽n−1} equipped with the inner product from the space L2(e−mQ(z)dA(z)). Here dA denotes a suitably normalized area measure on C. For a point z0 belonging to the interior of certain compact set S and satisfying ΔQ(z0)>0, we define the rescaled coordinatesz=z0+ξmΔQ(z0),w=z0+λmΔQ(z0). The following universality result is proved in the case q=2:1mΔQ(z0)|Kq,mQ,n(z,w)|e−12mQ(z)−12mQ(w)→|Lq−11(|ξ−λ|2)|e−12|ξ−λ|2 as m,n→∞ while n⩾m−M for any fixed M>0, uniformly for (ξ,λ) in compact subsets of C2. The notation Lq−11 stands for the associated Laguerre polynomial with parameter 1 and degree q−1. This generalizes a result of Ameur, Hedenmalm and Makarov concerning analytic polynomials to bianalytic polynomials. We also discuss how to generalize the result to q>2. Our methods include a simplification of a Bergman kernel expansion algorithm of Berman, Berndtsson and Sjöstrand in the one compex variable setting, and extension to the context of polyanalytic functions. We also study off-diagonal behaviour of the kernels Kq,mQ,n.