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arXiv · 2305.04354

Mean and variance of the cardinality of particles in polyanalytic Ginibre processes via a quantization method

Zouhaïr Mouayn·Mohamed Mahboubi·Othmane El Moize
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Abstract

We discuss the mean and variance of the number \textquotedblleft point-particles\textquotedblright\ $\sharp _{D_{R}}$\ inside a disk $D_{R}$ centered at the origin of the complex plane $\mathbb{C}$ and of radius $R>0$ with respect to a Ginibre-type (polyanalytic) process of index $m\in \mathbb{Z}_{+}$ by quantizing the phase space $\mathbb{C} $ via a set of generalized coherent states $\left\vert z,m\right\rangle $ of the harmonic oscillator on $L^{2}\left(\mathbb{R}\right) $. By this procedure, the spectrum of the quantum observable representing the indicator function $\chi _{D_{R}}$ of $ D_{R}$ (viewed as a classical observable) allows to compute the mean value of $\sharp _{D_{R}}$. The variance of $\sharp _{D_{R}}$ is obtained as a special eigenvalue of a quantum observable involving to the auto-convolution of $\chi _{D_{R}}.$ By adopting a coherent states quantization approach, we seek to identify classical observables on $\mathbb{C},$ whose quantum counterparts may encode the first cumulants of $\sharp _{D_{R}}$ through spectral properties.

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Zouhaïr Mouayn, Mohamed Mahboubi, Othmane El Moize. 2023-05-07. Mean and variance of the cardinality of particles in polyanalytic Ginibre processes via a quantization method. https://doi.org/10.1088/1742-5468%2Face0b4

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