The largest fragment of a homogeneous fragmentation process

TL;DR

This paper derives the asymptotic size of the largest fragment in a homogeneous fragmentation process, linking it to the process's Lévy exponent and confirming predictions from the theory of logarithmically correlated fields.

Contribution

It provides a precise asymptotic formula for the largest fragment size in homogeneous fragmentation processes, connecting probabilistic behavior with field classification theory.

Findings

Largest fragment size follows a specific exponential and polynomial decay formula.

Results align with predictions from the classification of homogeneous fragmentation as logarithmically correlated fields.

Offers a mathematical link between fragmentation process parameters and extreme fragment sizes.

Abstract

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.