The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s (2N)1 2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s>1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble an...
The eigenvalue probability density function ) is closely related to the Dyson Brownian motion model @cite_1 in random matrix theory (see Section 3 below). It is also referred to as a Gaussian ensemble with a source. In this context the case of @math having a finite rank has been studied by a number of authors @cite_26 @cite_21 @cite_10 @cite_35 . However our use of this differs in that we will keep @math fixed, and exhibit phase separation as a function of the perturbing parameter.
@cite_35 @cite_26 @cite_21 @cite_1 @cite_10
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Consider n non-intersecting particles on the real line (Dyson Brownian motions), all starting from the origin at time=0, and forced to return to x=0 at time=1. For large n, the average mean density of particles has its support, for each 0<t<1, within the interior of an ellipse. The Airy process is defined as the motion of these non-intersecting Brownian motions for large n, but viewed from an arbitrary point on the ellipse with an appropriate space-time rescaling. Assume now a finite number r of these particles are forced to a different target point. Does it affect the Brownian fluctuations along the ellipse for large n? In this paper, we show that no new process appears as long as one considers points on the ellipse, for which the t-coordinate is smaller than the t-coordinate of the point of tangency of the tangent to the curve passing through the target point. At this point of tangency the fluctuations obey a new statistics: the Airy process with r outliers (in short: r-Airy process ). The log of the transition probability of this new process is given by the Fredholm determinant of a new kernel (extending the Airy kernel) and it satisfies a non-linear PDE in x and the time. We continue the study of the Hermitian random matrix ensemble with external source Open image in new window where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a=1, since the eigenvalues of M accumulate on two intervals for a>1, and on one interval for 0 1 was treated in Part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0<a<1. As in Part I we apply the Deift Zhou steepest descent analysis to a 3×3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems. We present a random matrix interpretation of the distribution functions which have appeared in the study of the one-dimensional polynuclear growth (PNG) model with external sources. It is shown that the distribution, GOE @math , which is defined as the square of the GOE Tracy-Widom distribution, can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source, which have been studied in a different context previously. Compared to the original interpretation of the GOE @math as the square of GOE'', ours has an advantage that it can also describe the transition from the GUE Tracy-Widom distribution to the GOE @math . We further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain non-colliding Brownian motion model and the multi-layer PNG model with an external source. This provides us with a multi-matrix model interpretation of the multi-point height distributions of the PNG model with an external source. A new type of Coulomb gas is defined, consisting of n point charges executing Brownian motions under the influence of their mutual electrostatic repulsions. It is proved that this gas gives an exact mathematical description of the behavior of the eigenvalues of an (n × n) Hermitian matrix, when the elements of the matrix execute independent Brownian motions without mutual interaction. By a suitable choice of initial conditions, the Brownian motion leads to an ensemble of random matrices which is a good statistical model for the Hamiltonian of a complex system possessing approximate conservation laws. The development with time of the Coulomb gas represents the statistical behavior of the eigenvalues of a complex system as the strength of conservation‐destroying interactions is gradually increased. A virial theorem'' is proved for the Brownian‐motion gas, and various properties of the stationary Coulomb gas are deduced as corollaries. We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hard-edge of the spectrum of a random Hermitean matrix model. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann-Hilbert analysis of the corresponding orthogonal polynomials.