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Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians
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  • Published: 12 February 2014

Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians

  • Akihito Hora1 

Probability Theory and Related Fields volume 118, pages 115–130 (2000)Cite this article

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Abstract

On the adjacency algebra of a distance-regular graph we introduce an analogue of the Gibbs state depending on a parameter related to temperature of the graph. We discuss a scaling limit of the spectral distribution of the Laplacian on the graph with respect to the Gibbs state in the manner of central limit theorem in algebraic probability, where the volume of the graph goes to ∞ while the temperature tends to 0. In the model we discuss here (the Laplacian on the Johnson graph), the resulting limit distributions form a one parameter family beginning with an exponential distribution (which corresponds to the case of the vacuum state) and consisting of its deformations by a Bessel function.

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Authors and Affiliations

  1. Department of Environmental and Mathematical Sciences, Faculty of Environmental Science and Technology, Okayama University, Okayama 700-8530, Japan. e-mail: hora@math.ems.okayama-u.ac.jp, Japan

    Akihito Hora

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  1. Akihito Hora
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Additional information

Received: 7 July 1999 / Revised version: 23 February 2000 / Published online: 5 September 2000

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Cite this article

Hora, A. Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians. Probab Theory Relat Fields 118, 115–130 (2000). https://doi.org/10.1007/PL00008738

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  • Published: 12 February 2014

  • Issue Date: September 2000

  • DOI: https://doi.org/10.1007/PL00008738

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Keywords

  • Central Limit Theorem
  • Symmetric Group
  • Regular Graph
  • Spectral Distribution
  • Cayley Graph
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