  The model under study is an infinite 2D jellium of pointlike particles with
elementary charge $e$, interacting via the logarithmic potential and in thermal
equilibrium at the inverse temperature $\beta$. Two cases of the coupling
constant $\Gamma\equiv \beta e^2$ are considered: the Debye-H\"uckel limit
$\Gamma\to 0$ and the free-fermion point $\Gamma=2$. In the most general
formulation, two guest particles, the one with charge $q e$ (the valence $q$
being an arbitrary integer) and the hard core of radius $\sigma>0$ and the
pointlike one with elementary charge $e$, are immersed in the bulk of the
jellium at distance $d\ge \sigma$. Two problems are of interest: the asymptotic
large-distance behavior of the excess charge density induced in the jellium and
the effective interaction between the guest particles. Technically, the induced
charge density and the effective interaction are expressed in terms of
multi-particle correlations of the pure (translationally invariant) jellium
system. It is shown that the separation form of the induced charge density onto
its radial and angle parts, observed previously in the limit $\Gamma\to 0$, is
not reproduced at the coupling $\Gamma=2$. Based on an exact expression for the
effective interaction between guest particles at $\Gamma=2$, oppositely
($q=0,-1,-2,\ldots$) charged guest particles always attract one another while
likely ($q=1,2,\ldots$) charged guest particles repeal one another up to a
certain distance $d$ between them and then the mutual attraction takes place up
to an infinite distance.

Statistical Mechanics;Soft Condensed Matter