  The black-hole black-string system is known to exhibit critical dimensions
and therefore it is interesting to vary the spacetime dimension $D$, treating
it as a parameter of the system. We derive the large $D$ asymptotics of the
critical, i.e. marginally stable, string following an earlier numerical
analysis. For a background with an arbitrary compactification manifold we give
an expression for the critical mass of a corresponding black brane. This
expression is completely explicit for ${\bf T}^n$, the $n$ dimensional torus of
an arbitrary shape. An indication is given that by employing a higher
dimensional torus, rather than a single compact dimension, the total critical
dimension above which the nature of the black-brane black-hole phase transition
changes from sudden to smooth could be as low as $D\leq 11$.
