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Journal Articles Combinatorics, Probability and Computing Year : 2004

Additive decompositions, random allocations and threshold phenomena

Olivier Dubois
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Guy Louchard
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Jacques Mandler
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Abstract

An additive decomposition of a set $I$ of nonnegative integers is an expression of $I$ as the arithmetic sum of two other such sets. If the smaller of these has $p$ elements, we have a $p$-decomposition. If $I$ is obtained by randomly removing $n^{\alpha}$ integers from $\{0,\dots,n-1\}$, decomposability translates into a balls-and-urns problem, which we start to investigate (for large $n$) by first showing that the number of $p$-decompositions exhibits a threshold phenomenon as $\alpha$ crosses a $p$-dependent critical value. We then study in detail the distribution of the number of 2-decompositions. For this last case we show that the threshold is sharp and we establish the threshold function.

Dates and versions

hal-01185260 , version 1 (19-08-2015)

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Olivier Dubois, Guy Louchard, Jacques Mandler. Additive decompositions, random allocations and threshold phenomena. Combinatorics, Probability and Computing, 2004, 13 (4-5), pp.537-575. ⟨10.1017/S0963548304006224⟩. ⟨hal-01185260⟩
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