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It is well known that the mean field theory of directed polymers in a randommedium exhibits replica symmetry breaking with a distribution of overlaps whichconsists of two delta functions. Here we show that the leading finite sizecorrection to this distribution of overlaps has a universal character which canbe computed explicitly. Our results can also be interpreted as genealogicalproperties of branching Brownian motion or of branching random walks.

On the genealogy of branching random walks and of directed polymers