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Asymptotic Behaviour of Poisson-Dirichlet Distribution and Random Energy Model

  • Conference paper
XI Symposium on Probability and Stochastic Processes

Part of the book series: Progress in Probability ((PRPR,volume 69))

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Abstract

The family of Poisson-Dirichlet distributions is a collection of two-parameter probability distributions \(\{PD(\alpha,\theta ): 0 \leq \alpha < 1,\alpha +\theta > 0\}\) defined on the infinite-dimensional simplex. The parameters α and θ correspond to the stable and gamma component respectively. The distribution PD(α, 0) arises in the thermodynamic limit of the Gibbs measure of Derrida’s Random Energy Model (REM) in the low temperature regime. In this setting α can be written as the ratio between the temperature T and a critical temperature T c . In this paper, we study the asymptotic behaviour of PD(α, θ) as α converges to one or equivalently when the temperature approaches the critical value T c .

This work was supported by the Natural Sciences and Engineering Research Council of Canada.

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Acknowledgements

We wish to thank an anonymous referee for very helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada

    Shui Feng

  2. The School of Statistics and Mathematics, Zhongnan University of Economics and Law, 182 South Lake Avenue, East Lake New Technology Development Zone, Wuhan, 430073, China

    Youzhou Zhou

Authors
  1. Shui Feng
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  2. Youzhou Zhou
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Corresponding author

Correspondence to Shui Feng .

Editor information

Editors and Affiliations

  1. Probabilidad y Estadística, IIMAS UNAM, Mexico

    Ramsés H. Mena

  2. CIMAT, Guanajuato, Mexico

    Juan Carlos Pardo

  3. CIMAT, Guanajuato, Mexico

    Víctor Rivero

  4. Instituto de Matemáticas, UNAM, Mexico

    Gerónimo Uribe Bravo

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Feng, S., Zhou, Y. (2015). Asymptotic Behaviour of Poisson-Dirichlet Distribution and Random Energy Model. In: Mena, R., Pardo, J., Rivero, V., Uribe Bravo, G. (eds) XI Symposium on Probability and Stochastic Processes. Progress in Probability, vol 69. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-13984-5_7

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