On the second largest component of random hyperbolic graphs

We show that in the random hyperbolic graph model as formalized by Gugelmann et al. in the most interesting range of $\frac{1}{2}<\alpha <1$ the size of the second largest component is $\mathrm{\Theta }((\log n{)}^{1\mathrm{/}(1-\alpha )})$, thus answering a question of Bode et al. We also show that for $\alpha =\frac{1}{2}$ with constant probability the corresponding size is $\mathrm{\Theta }(\log n)$, whereas for $\alpha =1$ it is $\mathrm{\Omega }(n^b)$ for some $b>0$.