On the Largest Eigenvalue of a Hermitian Random Matrix Model with Spiked External Source II: Higher Rank Cases

Abstract

This is the second part of a study of the limiting distributions of the top eigenvalues of a Hermitian matrix model with spiked external source under a general external potential. The case when the external source is of rank one was analyzed in an earlier paper. In the present paper we extend the analysis to the higher rank case. If all the eigenvalues of the external source are less than a critical value, the largest eigenvalue converges to the right end-point of the support of the equilibrium measure as in the case when there is no external source. On the other hand, if an external source eigenvalue is larger than the critical value, then an eigenvalue is pulled off from the support of the equilibrium measure. This transition is continuous, and is universal, including the fluctuation laws, for convex potentials. For non-convex potentials, two types of discontinuous transitions are possible to occur generically. We evaluate the limiting distributions in each case for general potentials including those whose equilibrium measure have multiple intervals for their support.Here Z n is the normalization constant so that Hn p n (M )dM = 1 where dM denotes the Lebesgue measure. The sequence of probability spaces (H n , p n ), n = 1, 2, · · · , is called a Hermitian matrix model with external source matrices A n , n = 1, 2, · · · , and potential V . Note that due to the unitary invariance of dM and the presence of the trace in the exponent of (1), the density p n (M ) *

Random Matrices with Equispaced External Source

“…However, when the external field V (x) is general, the asymptotic analysis of the random matrix ensembles with external source has only had success for particular choices of external source matrices. Asymptotics for large n have been studied in [14,16,5,17,4,13,6] in the case where the external source matrix A has two different eigenvalues a and −a with equal multiplicity, and in [9,10,11,12] when A has a bounded, or slowly growing with n, number of non-zero eigenvalues. Large n asymptotics for general external source matrices have been considered in the physics literature, see e.g.…”

Limits of spiked random matrices II

“…It would be interesting to find natural "general β multi-spiked models" at finite n, interpolating between those studied here at β = 1, 2, 4 and generalizing those introduced in Part I for r = 1. At β = 2, perhaps one could discover a relationship with formulas of Baik and Wang (2013).…”

Eigenvector distribution in the critical regime of BBP transition

“…We also refer to [4], [7], [13], [14], [23], [30], [40], [45] and the reference therein for the first-order limit of the extreme eigenvalue of various related models. The fluctuation of the extreme eigenvalues of various models have been considered in [5], [4], [8], [9], [12], [17], [16], [19], [20], [25], [26], [31], [34], [40], [41], [45], [46], [47], [54], [55].…”