A critical dimension in the black-string phase transition

We consider black strings wrapped over the compact circle of a $d$-dimensional cylindrical spacetime. We construct static perturbative non-uniform string solutions around the Gregory-Laflamme point. First we compute the threshold instability mass for a large range of dimensions, $d$, and find that it follows essentially an exponential law $\gamma^d$, where $\gamma$ is a constant. Then we determine that there is a critical dimension, $d_*=13$, such that for $d\leq d_*$ the phase transition between the uniform and the non-uniform strings is first order, while for $d>d_*$, it is, surprisingly, higher order.