Are large distance Heegaard splittings generic?

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In a previous paper we introduced a notion of "genericity" for countable sets of curves in the curve complex of a surface S, based on the Lebesgue measure on the space of projective measured laminations in S. With this definition we prove that for each fixed g > 1 the set of irreducible genus g Heegaard splittings of high distance is generic, in the set of all irreducible Heegaard splittings. Our definition of "genericity" is different and more intrinsic then the one given via random walks.