  A new type of disorder-driven electronic percolation transition is found for
two-dimensional electron gas (2DEG), based on a quantum cellular automaton
model. This transition is shown to be accompanied with a metal-insulator
transition, as well as a singularity in the electronic compressibility. To this
end, the electronic system which is assumed to be in contact with an electronic
reservoir, is meshed by using of the phase coherence length $\zeta_{\phi}$ as
an extent which divides the spatial dynamics of the electrons into two separate
regimes and controls the localization of electrons. For the scales much smaller
than $\zeta_{\phi}$ the treatment is quantum mechanical, whereas for the scales
much larger than $\zeta_{\phi}$ the picture of semi-classical transport works
and the classical Mote Carlo method is used. Thomas-Fermi-Dirac (TFD) theory is
employed to find the dependence of the free energy of each cell on the
temperature ($T$), the (charged) disorder strength ($\Delta$) and the charge
content of the cell, and a cellular automaton model for transporting the
electrons between the cells is developed. At the transition line (in the
$T-\Delta$ space) the geometrical (e.g. correlation length) quantities also
diverge and some power-law behaviors emerge with some critical exponents in
agreement with the Gaussian free field (GFF) and the percolation theory. In the
percolating side the spanning cluster probability (SCP) (as a realization of
the conductivity) has a decreasing behavior in terms of the temperature which
is the characteristics of the metallic phase. Our simple model yields the
important features of the experimental observations, e.g. the singularity in
the conductivity in some critical density and also the universality
(non-universality) of the metal-insulator transition (MIT) for the small
(large) disorders in 2DEG.
