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

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.