Equivalence of ensembles under inhomogeneous conditioning and its applications to random Young diagrams

TL;DR

This paper proves the equivalence of ensembles for Bernoulli measures with inhomogeneous conditioning, extending local limit theorems, with applications to the study of random Young diagrams and their limiting Vershik curves.

Contribution

It extends classical local limit theorems to inhomogeneous Bernoulli sums and applies this to analyze random Young diagrams and their asymptotic shape.

Findings

Established ensemble equivalence under inhomogeneous conditioning

Extended local limit theorem for weighted Bernoulli sums

Connected results to the Vershik curve in Young diagram scaling limits

Abstract

We prove the equivalence of ensembles for Bernoulli measures on $\mathbb{Z}$ conditioned on two conserved quantities under the situation that one of them is spatially inhomogeneous. For the proof, we extend the classical local limit theorem for a sum of Bernoulli independent sequences to those multiplied by linearly growing weights. The motivation comes from the study of random Young diagrams. We discuss the relation between our result and the so-called Vershik curve which appears in a scaling limit for height functions of two-dimensional Young diagrams.