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Abstract: We consider the unitary matrix model in the limit where the size of thematrices become infinite and in the critical situation when a new spectral bandis about to emerge. In previous works the number of expected eigenvalues in aneighborhood of the band was fixed and finite, a situation that was termed-birth of a cut- or -first colonization-. We now consider the transitionalregime where this microscopic population in the new band grows without boundsbut at a slower rate than the size of the matrix. The local population in thenew band organizes in a -mesoscopic- regime, in between the macroscopicbehavior of the full system and the previously studied microscopic one. Themesoscopic colony may form a finite number of new bands, with a maximum numberdictated by the degree of criticality of the original potential. We describethe delicate scaling limit that realizes-controls the mesoscopic colony. Themethod we use is the steepest descent analysis of the Riemann-Hilbert problemthat is satisfied by the associated orthogonal polynomials.



Author: M. Bertola, S. Y. Lee, M. Y. Mo

Source: https://arxiv.org/



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