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

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Abstract

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.

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Acknowledgements

The author thanks Herbert Spohn for a lot of stimulating discussions for many years including the problems discussed in this paper. He also thanks Yoshiki Otobe for informing him the reference [5].

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  1. Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Tadahisa Funaki

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  1. Tadahisa Funaki
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Correspondence to Tadahisa Funaki.

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Dedicated to Herbert Spohn on the occasion of his 65th birthday.

Supported in part by the JSPS Grant (A) 22244007.

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Funaki, T. Equivalence of Ensembles Under Inhomogeneous Conditioning and Its Applications to Random Young Diagrams. J Stat Phys 154, 588–609 (2014). https://doi.org/10.1007/s10955-013-0841-6

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  • DOI: https://doi.org/10.1007/s10955-013-0841-6

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