  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 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 statistics. Namely, at small scales, the
point process still behaves 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 H^{1/2}- or H^1- correlated Gaussian noise 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.
