Incomplete Determinantal Processes: From Random Matrix to Poisson Statistics

We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or 2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of a log-gas confined in a general potential. We show that, depending on the expected number of deleted particles, there is a universal transition for mesoscopic linear statistics. Namely, at small scales, the point process behave according to random matrix theory, while, at large scales, it needs to be renormalized because the variance of any linear statistic diverges. The crossover is explicitly characterized as the superposition of a H1- or H1 / 2-correlated Gaussian noise depending on the dimension and an independent Poisson process. The proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process.