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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相空间和有限 Weyl-Heisenberg 系综中的调和分析


Weyl-Heisenberg 系综是与海森堡群的薛定谔表示相关的 $$\mathbb {R}^{2d}$$R2d 上的平移不变行列式点过程，并包括 Ginibre 系综和多分析系综作为示例，其模型更高的朗道物理学水平。我们引入了 Weyl-Heisenberg 系综的有限版本，并表明它们的行为与有限 Ginibre 系综类似。更具体地说，根据具有 N 个点的 Ginibre 系综渐近地接近无限 Ginibre 系综对面积 N 的圆盘的限制的观察，我们将有限 WH 系综定义为无限 WH 系综对给定域 $$\Omega $$Ω。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 膨胀并且过程按比例重新缩放（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 型多解析系综，该系综在单个 Landau 水平上对有限粒子系统进行建模，并用复杂的 Hermite 多项式定义。在技​​术层面上，我们证明有限 WH 系综为有限多解析 Ginibre 系综提供了近似模型，并且我们量化了相应的偏差。通过这种渐近描述，我们得出了热力学极限下多解析 Ginibre 系综的单点强度收敛速率的估计。