Boltzmann distribution on “short” integer partitions with power parts: Limit laws and sampling

TY - JOUR

T1 - Boltzmann distribution on “short” integer partitions with power parts

T2 - Limit laws and sampling

AU - Peyen, Jean

AU - Bogachev, Leonid V.

AU - Martin, Paul P.

N1 - Publisher Copyright: © 2024 The Author(s)

PY - 2024/8

Y1 - 2024/8

N2 - The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set Λˇ q of strict integer partitions (i.e., with unequal parts) into perfect q-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition weight (the sum of parts) and length (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters 〈N〉 and 〈M〉 controlling the expected weight and length, respectively. We study “short” partitions, where the parameter 〈M〉 is either fixed or grows slower than for typical partitions in Λˇ q. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed 〈M〉 and a limit shape result in the case of slow growth of 〈M〉. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyze their performance.

AB - The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set Λˇ q of strict integer partitions (i.e., with unequal parts) into perfect q-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition weight (the sum of parts) and length (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters 〈N〉 and 〈M〉 controlling the expected weight and length, respectively. We study “short” partitions, where the parameter 〈M〉 is either fixed or grows slower than for typical partitions in Λˇ q. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed 〈M〉 and a limit shape result in the case of slow growth of 〈M〉. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyze their performance.

KW - Integer partitions

KW - Boltzmann distribution

KW - Generating functions

KW - Young diagrams

KW - Limit shape

KW - Sampling algorithms

UR - https://www.scopus.com/pages/publications/85198739041

U2 - 10.1016/j.aam.2024.102739

DO - 10.1016/j.aam.2024.102739

M3 - Article

SN - 1090-2074

VL - 159

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

M1 - 102739

ER -