Abstract
For integers n,q=1,2,3,… , let Pol n,q denote the \({\mathbb{C}}\)-linear space of polynomials in z and \(\bar{z}\), of degree ≤n−1 in z and of degree ≤q−1 in \(\bar{z}\). 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→+∞ 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 \(\bar{\mathbb{D}}:=\{z\in{\mathbb{C}}:|z|\le1\}\). 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 \(\bar{\mathbb{D}}\). 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 \({\mathrm{d}}B^{\langle z\rangle}_{m,n,q}(w)\to{\mathrm{d}}\delta_{z} \) in the weak-star sense, where δ 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 \({\mathrm{d}}B^{\langle z\rangle}_{m,n,q}(w) \to{\mathrm{d}}\omega(w,z, {\mathbb{D}}^{e}) \), where \({\mathrm{d}}\omega(w,z,{\mathbb{D}}^{e})\) is the harmonic measure at z with respect to the exterior disk \({\mathbb{D}}^{e}:= \{w\in{\mathbb{C}}:\, |w|>1\}\). For boundary points, |z|=1, the Berezin measure \({\mathrm{d}}B^{\langle z\rangle}_{m,n,q}\) 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.
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Acknowledgements
We would like to thank Philip Kennerberg for sharing his MATLAB-code and discussing the practicalities in simulating the process. We also thank the referees for helpful suggestions.
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The second author is supported by the Göran Gustafsson Foundation (KVA) and Vetenskapsrådet (VR).
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Haimi, A., Hedenmalm, H. The Polyanalytic Ginibre Ensembles. J Stat Phys 153, 10–47 (2013). https://doi.org/10.1007/s10955-013-0813-x
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DOI: https://doi.org/10.1007/s10955-013-0813-x