Mesoscopic colonization of a spectral band
Abstract
We consider the unitary matrix model in the limit where the size of the matrices becomes infinite and in the critical situation when a new spectral band is about to emerge. In previous works, the number of expected eigenvalues in the 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 and controls the mesoscopic colony. The method we use is the steepest descent analysis of the RiemannHilbert problem that is satisfied by the associated orthogonal polynomials.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 October 2009
 DOI:
 10.1088/17518113/42/41/415204
 arXiv:
 arXiv:0904.1171
 Bibcode:
 2009JPhA...42O5204B
 Keywords:

 Mathematical Physics
 EPrint:
 17 pages, 2 figures, minor corrections and additions