[Paper Review] The rank 1 real Wishart spiked model I. Finite N analysis

M. Y. Mo|arXiv (Cornell University)|Nov 24, 2010

Random Matrices and Applications30 references3 citations

TL;DR

This paper establishes a finite-N analysis of the rank-1 real Wishart spiked model, deriving a Fredholm determinant formula for the largest eigenvalue distribution via orthogonal polynomial techniques. It provides the first exact computation of the asymptotic largest eigenvalue distribution at the phase transition point, differing from prior stochastic operator methods by using hyper-elliptic integrals and Zonal polynomial expansions on O(N).

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 $W_{\mathbb{R}}\left(Σ,M ight)$ in which all but one eigenvalues of $Σ$ is $1$. Let the non-trivial eigenvalue of $Σ$ be $1+τ$, then as $N$, $M ightarrow\infty$, with $N/M=γ^2$ finite and non-zero, the eigenvalue distribution of $S$ will converge into the Machenko-Pastur distribution inside a bulk region. As $τ$ 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 $\int_{O(N)}e^{- r(XgYg^T)}g^T\D g$ 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.

Motivation & Objective

• To analyze the finite-N behavior of the largest eigenvalue distribution in the rank-1 real Wishart spiked model.
• To derive an exact formula for the joint probability density function of eigenvalues using orthogonal polynomial techniques.
• To compute the asymptotics of the largest eigenvalue distribution at the phase transition point where a stray eigenvalue exits the bulk support.
• To provide a rigorous alternative to stochastic operator methods by using Zonal polynomial expansions and Haar integral evaluations on O(N).

Proposed method

• Derives a closed-form expression for the Haar integral ∫_{O(N)} exp(−tr(XgYg^T)) g^T dg when Y is rank-1 symmetric.
• Uses Zonal polynomials Z_p(X) to express the group integral as a series involving Z_{(k)}(X) and normalization constants Z_{(k)}(I_N).
• Applies generating function identities for Zonal polynomials to transform the integral into a contour integral over t.
• Evaluates the contour integral via residue at infinity, extracting the t^{-1} coefficient to obtain a hyper-elliptic integral representation.
• Establishes a Fredholm determinant formula for the largest eigenvalue distribution using the derived integral expression.
• Performs asymptotic analysis in the second half of the paper, focusing on the phase transition regime as N, M → ∞ with N/M = γ² fixed.

Experimental results

Research questions

• RQ1 How does the largest eigenvalue distribution behave in the finite-N regime for the rank-1 real Wishart spiked model?
• RQ2 What is the exact form of the joint eigenvalue probability density function for real Wishart matrices with a rank-1 covariance perturbation?
• RQ3 How can the Haar integral over O(N) be evaluated when the matrix Y is rank-1, enabling the computation of eigenvalue statistics?
• RQ4 What is the asymptotic behavior of the largest eigenvalue distribution at the phase transition point where a spike eigenvalue exits the bulk?
• RQ5 How does the orthogonal polynomial approach compare to stochastic operator methods in deriving the same asymptotic limit?

Key findings

• The paper derives an exact formula for the integral ∫_{O(N)} exp(−tr(XgYg^T)) g^T dg when Y is rank-1, expressed as a hyper-elliptic integral.
• The largest eigenvalue distribution is shown to be expressible via a Fredholm determinant involving orthogonal polynomials and Zonal polynomial expansions.
• The asymptotic analysis at the phase transition point is enabled by the finite-N formula, which allows the computation of the limiting distribution.
• The method avoids the use of stochastic operator techniques and instead relies on Zonal polynomial identities and residue calculus on contour integrals.
• The result provides the first exact finite-N derivation of the phase transition asymptotics for the real Wishart spiked model, resolving a gap left by previous works.
• The derivation confirms that the phase transition occurs precisely when the largest eigenvalue of S exits the support of the Marchenko-Pastur law, with a universal scaling limit.