Weyl–Heisenberg ensembles are translation-invariant determinantal point processes on $$\mathbb {R}^{2d}$$R2d associated with the Schrödinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl–Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with N points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area N, we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain $$\Omega $$Ω. We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of $$\Omega $$Ω, as $$\Omega $$Ω is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit.

中文翻译：

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相空间中的谐波分析和有限外尔-海森堡系综

我们为相应的单点强度收敛到 $$\Omega $$Ω 的指示函数提供了一个精确的速率，因为 $$\Omega $$Ω 是膨胀的并且该过程按比例重新调整（热力学状态）。构造和分析既不依赖于显式公式，也不依赖于正交多项式的渐近，而是依赖于相空间方法。其次，我们应用我们的构造来研究纯有限 Ginibre 型多分析系综，它在单个朗道水平上对有限粒子系统进行建模，并根据复 Hermite 多项式进行定义。在技​​术层面上，我们展示了有限 WH 集成为有限多分析 Ginibre 集成提供了一个近似模型，并且我们量化了相应的偏差。通过这种渐近描述，