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arXiv · 0904.1171

Mesoscopic colonization of a spectral band

M. Bertola·S. Y. Lee·M. Y. Mo
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Abstract

We consider the unitary matrix model in the limit where the size of the matrices become infinite and in the critical situation when a new spectral band is about to emerge. In previous works the number of expected eigenvalues in a neighborhood of the band was fixed and finite, a situation that was termed "birth of a cut" or "first colonization". We now consider the transitional regime where this microscopic population in the new band grows without bounds but at a slower rate than the size of the matrix. The local population in the new band organizes in a "mesoscopic" regime, in between the macroscopic behavior of the full system and the previously studied microscopic one. The mesoscopic colony may form a finite number of new bands, with a maximum number dictated by the degree of criticality of the original potential. We describe the delicate scaling limit that realizes/controls the mesoscopic colony. The method we use is the steepest descent analysis of the Riemann-Hilbert problem that is satisfied by the associated orthogonal polynomials.

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M. Bertola, S. Y. Lee, M. Y. Mo. 2009-04-16. Mesoscopic colonization of a spectral band. https://doi.org/10.1088/1751-8113%2F42%2F41%2F415204

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