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.
|Titolo:||Mesoscopic colonization of a spectral band|
|Autori:||Bertola M; Lee S Y; Mo M Y|
|Rivista:||JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL|
|Data di pubblicazione:||2009|
|Digital Object Identifier (DOI):||10.1088/1751-8113/42/41/415204|
|Appare nelle tipologie:||1.1 Journal article|