The rank 1 real Wishart spiked model

Abstract

In this paper, we consider N-dimensional real Wishart matrices Y in the class $W_{\mathbb{R}}(\Sigma ,M)$ in which all but one eigenvalues of $\Sigma$ is 1. Let the non-trivial eigenvalue of $\Sigma$ be $1+\tau$, then as N, $M\to \infty$, with $M/N={\gamma }^2$ finite and non-zero, the eigenvalue distribution of $Y$ will converge into the Marchenko-Pastur distribution inside a bulk region. When $\tau$ increases from zero, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko-Pastur density. As the this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest eigenvalue distribution of the Wishart matrix. In this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occur. We will first establish the results that are valid for all N and M and will use them to carry out the asymptotic analysis. In particular, we have derived a contour integral formula for the Harish-Chandra Itzykson-Zuber integral $\int_{O(N)}e^{-\tr(XgYg^T)}g^T\D g$ when X, Y are real symmetric and Y is a rank 1 matrix. This allows us to write down a Fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques. As a result, we obtain an integral formula for the largest eigenvalue distribution in the large N limit characterized by Painleve transcendents. The approach used in this paper is very different from a recent paper (Bloemendal and Virag ArXiv:1011.1877), in which the largest eigenvalue distribution was obtained using stochastic operator method. In particular, the Painleve formula for the largest eigenvalue distribution obtained in this paper is new.

Replacement of arXiv:1011.5404, asymptotic analysis included. 104 pages and 2 figures

References in corpus (16)

• Level-Spacing Distributions and the Airy Kernel
• Noise Dressing of Financial Correlation Matrices
• Fredholm Determinants, Differential Equations and Matrix Models
• Universal singularity at the closure of a gap in a random matrix theory
• Level Spacing of Random Matrices in an External Source
• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• Classical skew orthogonal polynomials and random matrices
• Limits of spiked random matrices I
• Random hermitian matrices in an external field
• Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions
• The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices
• The largest eigenvalues of sample covariance matrices for a spiked population: diagonal case
• The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I
• The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble
• On an average over the Gaussian Unitary Ensemble
• On Universality for Orthogonal Ensembles of Random Matrices

Cited by in corpus (5)

• Asymptotic power of sphericity tests for high-dimensional data
• Limits of spiked random matrices I
• The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I
• Three Problems Related to the Eigenvalues of Complex Non-central Wishart Matrices with a Rank-1 Mean
• Painleve II in random matrix theory and related fields