  We construct random point processes in the complex plane that are
asymptotically close to a given doubling measure. The processes we construct
are the zero sets of random entire functions that are constructed through
generalised Fock spaces. We offer two alternative constructions, one via bases
for these spaces and another via frames, and we show that for both
constructions the average distribution of the zero set is close to the given
doubling measure, and that the variance is much less than the variance of the
corresponding Poisson point process. We prove some asymptotic large deviation
estimates for these processes, which in particular allow us to estimate the
`hole probability', the probability that there are no zeroes in a given open
bounded subset of the plane. We also show that the `smooth linear statistics'
are asymptotically normal, under an additional regularity hypothesis on the
measure. These generalise previous results by Sodin and Tsirelson for the
Lebesgue measure.
