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We prove the equivalence of ensembles or a realization of the local equilibrium 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 and their evolutional models, which were originally suggested by Herbert Spohn. 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. We also discuss a related random dynamics.

Equivalence of Ensembles Under Inhomogeneous Conditioning and Its Applications to Random Young Diagrams