Ergodicity of multiplicative statistics

TL;DR

This paper establishes conditions under which rescaled Young diagrams from certain multiplicative measures on integer partitions converge to a deterministic limit shape, providing explicit formulas for the scaling and shape.

Contribution

It introduces new criteria for the convergence of Young diagrams to a limit shape in the context of multiplicative measures, including explicit formulas and new examples.

Findings

Rescaled Young diagrams converge to a deterministic limit shape under specified conditions.

Explicit formulas for the scaling function and limit shape are derived.

The results cover both known and new examples of multiplicative measures.

Abstract

For a subfamily of multiplicative measures on integer partitions we give conditions for properly rescaled associated Young diagrams to converge in probability to a certain deterministic curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape covering some known and some new examples.