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This paper is a continuation of a previous one [L. {\v{S}}amaj, {\it J. Stat.Phys.} {\bf 120}:125 (2005)] dealing with the renormalization of a guest chargeimmersed in a two-dimensional logarithmic Coulomb gas of pointlike $\pm$ unitcharges, the latter system being in the stability-against-collapse regime ofreduced inverse temperatures $0\le \beta <2$. In the previous work, using asine-Gordon representation of the Coulomb gas, an exact renormalized-chargeformula was derived for the special case of the {\em pointlike} guest charge$Q$, in its stability regime $\beta | Q| < 2$. In the present paper, we extendthe renormalized-charge treatment to the guest charge with a hard core ofradius $\sigma$, which allows us to go beyond the stability border $\beta| Q| =2$. In the limit of the hard-core radius much smaller than the correlationlength of the Coulomb-gas species and at a strictly finite temperature, due tothe counterion condensation in the extended region $\beta| Q| >2$, therenormalized charge $Q_{\rm ren}$ turns out to be a periodic function of thebare charge $Q$ with period 1. The renormalized charge therefore does notsaturate at a specific finite value as $| Q| \to\infty$, but oscillates betweentwo extreme values. In the high-temperature Poisson-Boltzmann scaling regime oflimits $\beta\to 0$ and $Q\to\infty$ with the product $\beta Q$ being finite,one reproduces correctly the monotonic dependence of $\beta Q_{\rm ren}$ on$\beta Q$ in the guest-charge stability region $\beta| Q| <2$ and theManning-Oosawa type of counterion condensation with the uniform saturation of$\beta Q_{\rm ren}$ at the value $4/\pi$ in the region $\beta| Q|\ge 2$.Comment: 28 pages, 1 figur

Renormalization of a Hard-Core Guest Charge Immersed in a Two-Dimensional Electrolyte