A Phase transition in the distribution of the length of integer partitions
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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 Erd˝os and Lehner published in 1941 that the distributions of the length in random restricted (d = 2) and random unrestricted (d 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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Paper presented at Strathmore International Math Research Conference on July 23 - 27, 2012
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 Erd˝os and Lehner published in 1941 that the distributions of the length in random restricted (d = 2) and random unrestricted (d 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.
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 Erd˝os and Lehner published in 1941 that the distributions of the length in random restricted (d = 2) and random unrestricted (d 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.
Keywords
Asymptotic expansions, Integer partitions, Multiplicities, Limit distribution