The largest fragment of a homogeneous fragmentation process

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We show that in homogeneous fragmentation processes the largest fragment at time t has size e-t Φ'(p)t-32 (Φ)'(p)+o(1), where Φ is the Lévy exponent of the fragmentation process, and p is the unique solution of the equation (Φ)'(p)=11+p. We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.