  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. 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.
