Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 104, 22 pages      arXiv:2106.14497      https://doi.org/10.3842/SIGMA.2021.104

Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

Masoumeh Koohestani a, Nobuaki Obata b and Hajime Tanaka b
a)  Department of Mathematics, K.N. Toosi University of Technology, Tehran 16765-3381, Iran
b)  Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan

Received July 19, 2021, in final form November 22, 2021; Published online November 26, 2021

Abstract
We determine the possible scaling limits in the quantum central limit theorem with respect to the Gibbs state, for a growing distance-regular graph that has so-called classical parameters with base unequal to one. We also describe explicitly the corresponding weak limits of the normalized spectral distribution of the adjacency matrix. We demonstrate our results with the known infinite families of distance-regular graphs having classical parameters and with unbounded diameter.

Key words: quantum probability; quantum central limit theorem; distance-regular graph; Gibbs state; classical parameters.

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