  Phase transitions and their associated crossovers are imprinted in the
behavior of fluctuations. Motivated by recent experiments on ultracold atoms in
optical lattices, we compute the thermodynamic density fluctuations $\delta
N^2$ of the two-dimensional fermionic Hubbard model with plaquette cellular
dynamical mean-field theory. To understand the length scale of these
fluctuations, we separate the local from the nonlocal contributions to $\delta
N^2$. We determine the effects of particle statistics, interaction strength
$U$, temperature $T$ and density $n$. At high temperature, our theoretical
framework reproduces the experimental observations in the doping-driven
crossover regime between metal and Mott insulator. At low temperature, there is
an increase of thermodynamic density fluctuations, analog to critical
opalescence, accompanied by a surprising reduction of the absolute value of
their nonlocal contributions. This is a precursory sign of an underlying phase
transition between a pseudogap phase and a metallic phase in doped Mott
insulators, which should play an important role in the cuprate high-temperature
superconductors. Predictions for ultracold atom experiments are made.
