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

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.