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

Abstract

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers [1,20] and sample covariance matrices [4]. We consider the case when the n × n external source matrix has two distinct real eigenvalues: a with multiplicity r and zero with multiplicity n − r. The source is small in the sense that r is finite or r = O(n γ ), for 0 < γ < 1. For a Gaussian potential, Péché [29] showed that for |a| sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for |a| sufficiently large (the supercritical regime) r eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of r × r Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.

Laguerre polynomials and transitional asymptotics of the modified Korteweg–de Vries equation for step-like initial data

“…Statements of the type of Lemma 5.2 fall within the general class of "normal form" of singularities. A completely similar lemma can be found in [2] (Prop. 2.2) with the only difference that in our case z(y; t) = O(y) uniformly as t → ∞ while in the case of [2] we have O(y 2 ) and hence the normal form starts with z 2 in the right side of (98).…”

“…A completely similar lemma can be found in [2] (Prop. 2.2) with the only difference that in our case z(y; t) = O(y) uniformly as t → ∞ while in the case of [2] we have O(y 2 ) and hence the normal form starts with z 2 in the right side of (98). We omit the detailed proof because it is not substantially different from the one referred to above.…”

“…Similar phenomena occur in the semiclassical asymptotics of integrable equations and large size random matrices, see [4], [2], [3]. The so-called Riemann-Hilbert method (a.k.a.…”

“…Theorem 1. [30] In the region −6c 2 t < x < 4c 2 t the solution of the initial value problem (1), (2), (3) takes form of a modulated elliptic wave q(x, t) = q ell (x, t) + o (1) , t → ∞, where q ell (x, t) = c 2 − d 2 (ξ) Θ (πi + itB(d(ξ)) + i∆(d(ξ))|τ (d(ξ))) Θ (itB(d(ξ)) + i∆(d(ξ))|τ (d(ξ))) , ξ = x 12t .…”

Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices

“…For certain special choices of M n , it is known that other limiting kernels can appear. If M n has only two distinct eigenvalues, the Pearcey kernel arises at a critical time [14,13,43,10,3], and generalizations of the Airy kernel can appear at the edge [1,2,7,8]. Some of these results have been obtained using a representation of the eigenvalue correlation kernel (see (1.9) below) in terms of multiple orthogonal polynomials.…”