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 «point-particles» $\sharp_{D_R}$ inside a disk $D_R$ centered at the origin of the complex plane $C$ and of radius $R>0$ with respect to a Ginibre-type (polyanalytic) process of index $m$ in $Z+$ by quantizing the phase space $C$ via a set of generalized coherent states $|z,m>$ of the harmonic oscillator on $L^2(R)$. By this procedure, the spectrum of the quantum observable representing the indicator function $\chi_{D_R}$ (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 $C$, whose quantum counterparts may encode the first cumulants of $\sharp_{D_R}$ through spectral properties.
This is a joint work with Mohamed Mahboubi and Othmane El Moize.