Renormalization of a hard-core guest charge immersed in a two-dimensional electrolyte

This paper is a continuation of a previous one [L. Samaj, J. Stat. Phys. 120: 125 (2005)] dealing with the renormalization of a guest charge immersed in a two-dimensional logarithmic Coulomb gas of pointlike +/- unit charges, the latter system being in the stability-against-collapse regime of reduced inverse temperatures 0 <= beta < 2. In the previous work, using a sine-Gordon representation of the Coulomb gas, an exact renormalized-charge formula was derived for the special case of the pointlike guest charge Q, in its stability regime beta vertical bar Q vertical bar < 2. In the present paper, we extend the renormalized-charge treatment to the guest charge with a hard core of radius sigma, which allows us to go beyond the stability border beta vertical bar Q vertical bar = 2. In the limit of the hard-core radius much smaller than the correlation length of the Coulomb-gas species and at a strictly finite temperature, due to the counterion condensation in the extended region beta vertical bar Q vertical bar > 2, the renormalized charge Q(ren) turns out to be a periodic function of the bare charge Q with period 1. The renormalized charge therefore does not saturate at a specific finite value as vertical bar Q vertical bar --> infinity, but oscillates between two extreme values. In the high-temperature Poisson-Boltzmann scaling regime of limits beta --> 0 and Q --> infinity with the product beta Q being finite, one reproduces the Manning-Oosawa type of counterion condensation with the uniform saturation of beta Q(ren) at the value 4/pi in the region beta vertical bar Q vertical bar >= 2. The obtained results disprove the "regularization hypothesis" of the previous work about the possibility of an analytic continuation of the formula for Q(ren) from the stability region beta vertical bar Q vertical bar < 2 to beta vertical bar Q vertical bar >= 2.