The largest fragment of a homogeneous fragmentation process

We show that in homogeneous fragmentation processes the largest fragment at time $t$ has size $e^{-t \Phi’(\bar{p})}t^{-\frac32 (\log \Phi)'(\bar{p})+o(1)},$where$\Phi$ is the L'evy exponent of the fragmentation process, and $\bar{p}$ is the unique solution of the equation $(\log \mathrm{\Phi }{)}^{\mathrm{\prime }}(\bar{p})=\frac{1}{1+\bar{p}}$. We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.