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 Levy 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.