Mean and variance of the cardinality of particles in infinite true polyanalytic Ginibre processes via a coherent states quantization method

We discuss the mean and variance of the number ‘point-particles’ ♯DR inside a disk D R centered at the origin of the complex plane C and of radius R > 0 with respect to an infinite true polyanalytic process of index m∈Z+ by quantizing the phase space C via a set of generalized coherent states (CSs) z,m of the harmonic oscillator on L2R . By this procedure, the spectrum of the quantum observable representing the indicator function χDR ofD R (viewed as a classical observable) allows to compute the mean value of ♯DR . The variance of ♯DR is obtained as a special eigenvalue of a quantum observable involving the auto-convolution of χDR . By adopting a CSs quantization approach, we seek to identify classical observables on C, whose quantum counterparts may encode the first cumulants of ♯DR through spectral properties.

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