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

We discuss the mean and variance of the number \textquotedblleft point-particles\textquotedblright\ ${\mathrm{♯}}_{D_R}$\ 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 $\mid z,m⟩$ 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 ${\mathrm{♯}}_{D_R}$. The variance of ${\mathrm{♯}}_{D_R}$ is obtained as a special eigenvalue of a quantum observable involving to the auto-convolution of ${\chi }_{D_R}\mathrm{.}$ 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 ${\mathrm{♯}}_{D_R}$ through spectral properties.