Mesoscopic colonization of a spectral band

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.

Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

“…This implies that we can replace ζ x and ζ y in the kernel by the leading approximation in (6-23) without changing the error in the kernel. The mismatch from this approximation produces an error of smaller order than the one already indicated in (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) and yields the overall error term O(n −1/3 ) advocated in Theorem 1.…”

“…where the middle term of (5)(6)(7)(8)(9)(10)(11)(12)(13)(14) is hidden in O( ). Because det H(z; n) = det H 0 = 1, the big matrix of size qk×qk multiplied on the right of [F 1 , ..., F k ] is invertible and, therefore, the solution can be uniquely obtained.…”

“…Furthermore, using (2)(3)(4) and (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12), We label these basis elements as N j with j = 1, . .…”

“…Lemma 5.6. In the near-critical regime, for large n, For part (c), first recall from the Schlesinger calculations that F (1,m) = O(r/n 1/2 ) and F (2,m) = O(r/n 1/3 ) (as follows from (5)(6)(7)(8)(9)(10)(11), (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22), and ). Recalling the definition of R(z) in , we see that…”

“…, (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24) and evaluating K(x, y) at…”