  Multi-point probability measures along with the dielectric function of Dirac
Fermions in mono-layer graphene containing particle-particle and white-noise
(out-plane) disorder interactions on an equal footing in the Thomas-Fermi-Dirac
approximation is investigated. By calculating the one-body carrier density
probability measure of the graphene sheet, we show that the density fluctuation
($\zeta^{-1}$) is related to the disorder strength ($n_i$), the interaction
parameter ($r_s$) and the average density ($\bar{n}$) via the relation
$\zeta^{-1}\propto r_sn_i^2\bar{n}^{-1}$ for which $\bar{n}\rightarrow 0$ leads
to strong density inhomogeneities, i.e. electron-hole puddles (EHPs), in
agreement with the previous works. The general equation governing the two-body
distribution probability is obtained and analyzed. We present the analytical
solution for some limits which is used for calculating density-density response
function. We show that the resulting function shows power-law behaviors in
terms of $\zeta$ with fractional exponents which are reported. The
disorder-averaged polarization operator is shown to be a decreasing function of
momentum like ordinary 2D parabolic band systems. It is seen that a
disorder-driven momentum $q_{\text{ch}}$ emerges in the system which controls
the behaviors of the screened potential. We show that in small densities an
instability occurs in which imaginary part of the dielectric function becomes
negative and the screened potential changes sign. Corresponding to this
instability, some oscillations in charge density along with a
screening-anti-screening transition are observed. These effects become dominant
in very low densities, strong disorders and strong interactions, the state in
which EHPs appear. The total charge probability measure is another quantity
which has been investigated in this paper.
