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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Authors: | Akihito Hora |
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Institution: | (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, JP |
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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.
Received: 7 July 1999 / Revised version: 23 February 2000 / Published online: 5 September 2000 |
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