Abstract
We compute the first twenty moments of three convergent quartic bi-tracial 2-matrix ensembles in the large N limit. These ensembles are toy models for Euclidean quantum gravity originally proposed by John Barrett and collaborators. A perturbative solution is found for the first twenty moments using the Schwinger-Dyson equations and properties of certain bi-colored unstable maps associated to the model. We then apply a result of Guionnet et al. to show that the perturbative and convergent solution coincide for a small neighbourhood of the coupling constants. For each model we compute an explicit expression for the free energy, critical points, and critical exponents in the large N limit. In particular, the string susceptibility is found to be γ = 1/2, hinting that the associated universality class of the model is the continuous random tree.
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Acknowledgments
We would like to thank Hamed Hessam for the long discussions and hard work that preceded these results. We are also grateful for funding from the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Khalkhali, M., Pagliaroli, N. Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry. J. High Energ. Phys. 2024, 186 (2024). https://doi.org/10.1007/JHEP05(2024)186
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DOI: https://doi.org/10.1007/JHEP05(2024)186