A Hierarchical Model of Quantum Anharmonic Oscillators: Critical Point Convergence |
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Authors: | Sergio Albeverio Agnieszka Kozak Yuri Kozitsky |
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Institution: | (1) Istituto Nazionale di Fisica Nucleare, Sezione di Torino and Dipartimento di Fisica Teorica dell Università di Torino, via P. Giuria 1, 10125 Torino, Italy;(2) Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, 15100 Alessandria, Italy |
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Abstract: | We study the two-dimensional gauge theory of the symmetric group Sn describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disc and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of Sn, one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra logN compared to the standard t Hooft scaling. |
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