TY - JOUR

T1 - Growing integer partitions with uniform marginals and the equivalence of partition ensembles

AU - Якубович, Юрий Владимирович

PY - 2024/11/1

Y1 - 2024/11/1

N2 - 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.

AB - 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.

KW - Equivalence of ensembles

KW - Even parts

KW - Integer partition

KW - Limit shape

KW - Odd parts

KW - Random growth process

UR - https://www.mendeley.com/catalogue/fe606f7e-4234-37b7-9fba-ae1d322909d8/

U2 - 10.1016/j.aim.2024.109908

DO - 10.1016/j.aim.2024.109908

M3 - Article

VL - 457

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 109908

ER -