  We numerically construct static localized black holes in five and six
spacetime dimensions which are solutions to Einstein's vacuum field equations
with one compact periodic dimension. In particular, we investigate the critical
regime in which the poles of the localized black hole are about to merge. A
well adapted multi-domain pseudo-spectral scheme provides us with accurate
results and enables us to investigate the phase diagram of those localized
solutions within the critical regime, which goes far beyond previous results.
We find that in this regime the phase diagram possesses a spiral structure
adapting to the one recently found for non-uniform black strings. When
approaching the common endpoint of both phases, the behavior of physical
quantities is described by complex critical exponents giving rise to a discrete
scaling symmetry. The numerically obtained values of the critical exponents
agree remarkably well with those derived from the double-cone metric.
