The rank 1 real Wishart spiked model I. Finite N analysis

Abstract

This is the first part of a paper that studies the phase transition in the asymptotic limit of the rank 1 real Wishart spiked model. In this paper, we consider $N$-dimensional real Wishart matrices $S$ in the class $W_{\mathbb{R}}\left(\Sigma,M\right)$ 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 $N/M=\gamma^2$ finite and non-zero, the eigenvalue distribution of $S$ will converge into the Machenko-Pastur distribution inside a bulk region. As $\tau$ increases from zero, one starts seeing stray eigenvalues of $S$ outside of the support of the Machenko-Pastur density. As the first of these stray eigenvalues leaves the bulk region, a phase transition will occur in the largest eigenvalue distribution of the Wishart matrix. In this paper will compute the asymptotics of the largest eigenvalue distribution when the phase transition occur. In the this first half of the paper, we will establish the results that are valid for all $N$ and $M$ and will use them to carry out the asymptotic analysis in the second half of the paper, which will follow shortly. In particular, we have derived a formula for the integral $\int_{O(N)}e^{-\tr(XgYg^T)}g^T\D g$ when $X$, $Y$ are 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. This approach is very different from a recent paper by Bloemendal and Virag, in which the largest eigenvalue distribution was obtained using stochastic operator method.