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 $(\log \Phi)'(\bar{p})=\frac1{1+\bar{p}}$. We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.