The rank 1 real Wishart spiked model

Understand

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, $M\rightarrow\infty$, 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.