  We address the critical and universal aspects of counterion-condensation
transition at a single charged cylinder in both two and three spatial
dimensions using numerical and analytical methods. By introducing a novel
Monte-Carlo sampling method in logarithmic radial scale, we are able to
numerically simulate the critical limit of infinite system size (corresponding
to infinite-dilution limit) within tractable equilibration times. The critical
exponents are determined for the inverse moments of the counterionic density
profile (which play the role of the order parameters and represent the inverse
localization length of counterions) both within mean-field theory and within
Monte-Carlo simulations. In three dimensions (3D), correlation effects
(neglected within mean-field theory) lead to an excessive accumulation of
counterions near the charged cylinder below the critical temperature
(condensation phase), while surprisingly, the critical region exhibits
universal critical exponents in accord with the mean-field theory. In two
dimensions (2D), we demonstrate, using both numerical and analytical
approaches, that the mean-field theory becomes exact at all temperatures
(Manning parameters), when number of counterions tends to infinity. For finite
particle number, however, the 2D problem displays a series of peculiar singular
points (with diverging heat capacity), which reflect successive de-localization
events of individual counterions from the central cylinder. In both 2D and 3D,
the heat capacity shows a universal jump at the critical point, and the energy
develops a pronounced peak. The asymptotic behavior of the energy peak location
is used to locate the critical temperature, which is also found to be universal
and in accordance with the mean-field prediction.
