# A phase transition in the distribution of the length of integer partitions

Abstract : We assign a uniform probability to the set consisting of partitions of a positive integer $n$ such that the multiplicity of each summand is less than a given number $d$ and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erdős and Lehner published in 1941 that the distributions of the length in random restricted $(d=2)$ and random unrestricted $(d \geq n+1)$ partitions behave very differently. In this paper we show that as the bound $d$ increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case.
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Dimbinaina Ralaivaosaona. A phase transition in the distribution of the length of integer partitions. 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'12), 2012, Montreal, Canada. pp.265-282, ⟨10.46298/dmtcs.2999⟩. ⟨hal-01197255⟩

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