We present an explicit construction of a Markovian random growth process on integer partitions such that given it visits some level n, it passes through any partition λ of n with equal probabilities. The construction has continuous time, but we also investigate its discrete time jump chain. The jump probabilities are given by explicit but complicated expressions, so we find their asymptotic behavior as the partition becomes large. This allows us to explain how the limit shape is formed. Using the known connection of the considered probabilistic objects to Poisson point processes, we give an alternative description of the partition growth process in these terms. Then we apply the constructed growth process to find sufficient conditions for a phenomenon known as equivalence of two ensembles of random partitions for a finite number of partition characteristics. This result allows to show that counts of odd and even parts in a random partition of n are asymptotically independent as n→∞ and to find their limiting distributions, which are, somewhat surprisingly, different.
Original languageEnglish
Article number109908
Number of pages41
JournalAdvances in Mathematics
Volume457
DOIs
StatePublished - 1 Nov 2024

    Research areas

  • Equivalence of ensembles, Even parts, Integer partition, Limit shape, Odd parts, Random growth process

    Scopus subject areas

  • Statistics and Probability

ID: 124222216