The largest fragment of a homogeneous fragmentation process

Abstract

We show that in homogeneous fragmentation processes the largest fragment at time $t$ has size $e^{-t{\Phi }^{\prime }(\bar{p})}{t}^{-\frac{3}{2}(\log \Phi {)}^{\prime }(\bar{p})+o(1)},$ where $\Phi$ is the Lévy exponent of the fragmentation process, and $\bar{p}$ is the unique solution of the equation $(\log \Phi {)}^{\prime }(\bar{p})=\frac{1}{1+\bar{p}}$. We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.

20 pages

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