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For systems broken in pieces, an immediate observable is the size of the largest fragment. Despite the apparent simplicity of this variable, one can extract de finite information about the correlations present in the system, and even the mechanisms which led to this special fragment. The underlying idea is that the largest fragment is indeed very specia l, because, during the history of its formation, this fragment avoided (to some extent) processes of fragmentation and most of the relevant correlations. However, to get the useful informat ion, average values are not enough, and one has to investigate in details the complete fluctuations o f the largest-fragment size, under the form of its probability-distribution function. Focusing h ere on principles and not on particular applications, we take two simple models of aggregation exhibiting critical behavior (at-equilibrium percolation and dynamical Smoluchowski equations), we show and discuss in the present work the way the information is hidden in the largest-fragment distribution. Moreover, the distribution of the second largest fragment gives precise insights about the mechanisms which generated the system. This approach provides a novel kind of data analysis, which can be helpful when one has only access to the final state of a system of fragments, as it ma y happen in nuclear collisions or in explosion of hot atomic clusters.

Where are the correlations hidden in the distribution of the largest fragment?