Rank 1 real Wishart spiked model

Abstract

In this paper, we consider N -dimensional real Wishart matrices Y in the class W R (Σ, M ) in which all but one eigenvalues of Σ is 1. Let the non-trivial eigenvalue of Σ be 1 + τ , then as N , M → ∞, with M/N = γ 2 finite and non-zero, the eigenvalue distribution of Y will converge into the Marchenko-Pastur distribution inside a bulk region. When τ 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 O(N ) e tr(XgY g T ) g T dg 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 Painlevé transcendents. The approach used in this paper is very different from a recent paper [23], in which the largest eigenvalue distribution was obtained using stochastic operator method. In particular, the Painlevé formula for the largest eigenvalue distribution obtained in this paper is new.

Painlevé II in Random Matrix Theory and Related Fields

“…The significance of the coupling implied by (2.40) and (2.43) is that it corresponds to a scaling about this critical value, while the scaling of x 1 centres about the leading order value of the largest eigenvalue, and measures in units such that the largest eigenvalues have spacing of order unity. The description of the rank 1 perturbed scaled soft edge state for β-ensembles ((2.40) corresponds to β = 2; (2.43) to β = 4) has been the subject of a number of recent papers [10,67,28,51].…”

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR Β-Ensembles

“…The derivation relies on an unproved conjecture relating to the asymptotics of a certain generalized hypergeometric function. Such multivariable special functions, introduced into random matrix theory by Constantine and Muirhead in the case β = 1 in the 60's and 70's (see [31]), are finding their way into a number of recent works relating to asymptotics of eigenvalue distributions [7,36,29,18,8,19]. The result of [22] also assumes knowledge of the corresponding asymptotics of E hard β (0; (0, s); βa/2).…”

Joint density of eigenvalues in spiked multivariate models

“…Particular cases of are already known: 0 F 0 for both real and complex cases , for general α , and , and for the complex case only, 0 F 1 and 1 F 1 . also gives formula in the 0 F 0 case.…”