The Size of the Largest Part of Random Weighted Partitions of Large Integers

TL;DR

This paper proves that the largest part in a randomly chosen weighted partition of a large integer converges to a Gumbel distribution under classical assumptions, extending known results in partition theory and statistical physics.

Contribution

It extends classical limit theorems for integer partitions to weighted partitions under Meinardus' assumptions, revealing the asymptotic distribution of the largest part.

Findings

Largest part converges to Gumbel distribution as n increases

Extends known results to weighted partitions under classical assumptions

Connects partition theory with statistical physics models

Abstract

For a given sequence of weights (non-negative numbers), we consider partitions of the positive integer n. Each n-partition is selected uniformly at random from the set of all such partitions. Under a classical scheme of assumptions on the weight sequence, which are due to Meinardus (1954), we show that the largest part in a random weighted partition, appropriately normalized, converges weakly, as n tends to infinity, to a random variable having the extreme value (Gumbel's) distribution. This limit theorem extends some known results on particular types of integer partitions and on the Bose-Einstein model of ideal gas.