For integers n,q=1,2,3,aEuro broken vertical bar aEuro parts per thousand, let Pol (n,q) denote the -linear space of polynomials in z and , of degree a parts per thousand currency signn-1 in z and of degree a parts per thousand currency signq-1 in . We supply Pol (n,q) with the inner product structure of the resulting Hilbert space is denoted by Pol (m,n,q) . Here, it is assumed that m is a positive real. We let K (m,n,q) denote the reproducing kernel of Pol (m,n,q) , and study the associated determinantal process, in the limit as m,n ->+a while n=m+O(1); the number q, the degree of polyanalyticity, is kept fixed. We call these processes polyanalytic Ginibre ensembles, because they generalize the Ginibre ensemble-the eigenvalue process of random (normal) matrices with Gaussian weight. There is a physical interpretation in terms of a system of free fermions in a uniform magnetic field so that a fixed number of the first Landau levels have been filled. We consider local blow-ups of the polyanalytic Ginibre ensembles around points in the spectral droplet, which is here the closed unit disk . We obtain asymptotics for the blow-up process, using a blow-up to characteristic distance m (-1/2); the typical distance is the same both for interior and for boundary points of . This amounts to obtaining the asymptotical behavior of the generating kernel K (m,n,q) . Following (Ameur et al. in Commun. Pure Appl. Math. 63(12):1533-1584, 2010), the asymptotics of the K (m,n,q) are rather conveniently expressed in terms of the Berezin measure (and density) For interior points |z|< 1, we obtain that in the weak-star sense, where delta (z) denotes the unit point mass at z. Moreover, if we blow up to the scale of m (-1/2) around z, we get convergence to a measure which is Gaussian for q=1, but exhibits more complicated Fresnel zone behavior for q > 1. In contrast, for exterior points |z|> 1, we have instead that , where is the harmonic measure at z with respect to the exterior disk . For boundary points, |z|=1, the Berezin measure converges to the unit point mass at z, as with interior points, but the blow-up to the scale m (-1/2) exhibits quite different behavior at boundary points compared with interior points. We also obtain the asymptotic boundary behavior of the 1-point function at the coarser local scale q (1/2) m (-1/2).
2013. Vol. 153, no 1, 10-47 p.