Global fluctuations for standard Young tableaux

TL;DR

This paper develops a framework using Young generating functions to analyze probability measures on integer partitions, characterizing their fluctuations and establishing a multilevel CLT for studying random standard Young tableaux, with fluctuations converging to a conditioned Gaussian Free Field.

Contribution

It introduces Young generating functions for probability measures on partitions and proves a multilevel central limit theorem for random Young tableaux, linking fluctuations to Gaussian Free Fields.

Findings

Fluctuations of height functions converge to a conditioned Gaussian Free Field.

Characterization of distributions satisfying law of large numbers and CLT.

Application to Plancherel growth process and infinite symmetric group characters.

Abstract

We introduce the notion of a Young generating function for a probability measure on integer partitions. We use this object to characterize probability distributions over integer partitions satisfying a law of large numbers and those that satisfy a central limit theorem. We further establish a multilevel central limit theorem, which enables the study of random standard Young tableaux. As applications of these results, we describe the fluctuations of height functions associated with (i) the Plancherel growth process, (ii) random standard Young tableaux of fixed shape, and (iii) probability distributions induced by extreme characters of the infinite symmetric group $S_\infty$. In all cases, we identify the limiting fluctuations as a conditioned Gaussian Free Field.