  We test the concepts of renormalized charge and potential saturation,
introduced within the framework of highly asymmetric Coulomb mixtures, on
exactly solvable Coulomb models. The object of study is the average
electrostatic potential induced by a unique "guest" charge immersed in a
classical electrolyte, the whole system being in thermal equilibrium at some
inverse temperature $\beta$. The guest charge is considered to be either an
infinite hard wall carrying a uniform surface charge or a charged colloidal
particle. The systems are treated as two-dimensional; the electrolyte is
modelled by a symmetric two-component plasma of point-like $\pm e$ charges with
logarithmic Coulomb interactions. Two cases are solved exactly: the
Debye-H\"uckel limit $\beta e^2\to 0$ and the Thirring free-fermion point
$\beta e^2 = 2$. The results at the free-fermion point can be summarized as
follows: (i) The induced electrostatic potential exhibits the asymptotic
behavior, at large distances from the guest charge, whose form is different
from that obtained in the Debye-H\"uckel (linear Poisson-Boltzmann) theory.
This means that the concept of renormalized charge, developed within the
nonlinear Poisson-Boltzmann theory to describe the screening effect of the
electrolyte cloud, fails at the free-fermion point. (ii) In the limit of an
infinite bare charge, the induced electrostatic potential saturates at a finite
value in every point of the electrolyte region. This fact confirms the
previously proposed hypothesis of potential saturation.
