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Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. (English) Zbl 1086.15022

The largest eigenvalue distribution is investigated for $N$-dimensional random Gram matrices (“covariance matrices”) $S={M}^{-1}\left(X-\overline{X}\right){\left(X-\overline{X}\right)}^{T}$, where $X$ are rectangular $N×M$ sample matrices whose columns are observation vectors distributed as $N\left(0,{\Sigma }\right)$, and $\overline{X}$ are sample mean vectors. It is assumed that $N,M\to \infty$ and $M/N\to {\gamma }^{2}>1$. In this asymptotics, when ${\Sigma }$ are unity matrices, V. A. Marchenko and L. A. Pastur [Math. USSR, Sb. 1, 457–483 (1967; Zbl 0162.22501)] derived well-known limiting distribution density of eigenvalues of $S$; its upper bound was found to be ${\left(1+1/\gamma \right)}^{2}$. For this case also, the largest eigenvalue ${\lambda }_{1}$ of the limiting distribution was found by P. J. Forrester [Nucl. Phys. B 402, No. 3, 709–728 (1993; Zbl 1043.82538)]

$ℙ\left(a{M}^{2/3}\left({\lambda }_{1}-{\left(1+1/\gamma \right)}^{2}

where $a$ depends only on $\gamma$.

The authors find the limiting distribution of the largest eigenvalue ${\lambda }_{1}$ for complex matrices $S$ when there is a finite number of eigenvalues of ${\Sigma }$ different from 1. Let $r\le N$ and the eigenvalues ${l}_{1}\ge {l}_{2}\ge \cdots {l}_{N}$ of ${\Sigma }$ be such that ${l}_{r+1}={l}_{r+2}=\cdots ={l}_{N}·$

Theorem 1 states that

(i) If $k\le r$, ${l}_{1}={l}_{2}=\cdots ={l}_{k}=1+1/\gamma$ and all ${\lambda }_{j}$, $j=k+1,k+2\cdots ,r$, are located on a compact within $\left(0,1+1/\gamma \right)$, then the largest eigenvalue ${\lambda }_{1}$ of $S$ has the same limiting distribution (1);

(ii) If for some $k\le r$, the eigenvalues ${l}_{1}={l}_{2}=\cdots ={l}_{k}$ are located on a compact within $\left(1+1/\gamma ,\infty \right)$ and all eigenvalues ${l}_{j}$, $j=k+1,k+2\cdots ,r$, are on a compact within $\left(0,{l}_{1}\right)$, then

$ℙ\left(a{M}^{1/2}\left({\lambda }_{1}-c\right)\le x\right)\to {G}_{k}\left(x\right),$

where $a>0$ and $c<{l}_{1}$ depend on $\gamma$ and ${l}_{1}$, and ${G}_{k}\left(x\right)$ is the distribution function for the largest eigenvalue of $k×k$ random matrix when ${\Sigma }$ is unity.

Theorem 2 establishes the limiting distribution (1) in an intermediate case. Two applications are considered: finding the last passage time in a percolation problem and a queueing model of a sequential service of $M$ customers in the line of $N$ tellers under a fixed order of serving.

##### MSC:
 15A52 Random matrices (MSC2000) 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 62E20 Asymptotic distribution theory in statistics 15A18 Eigenvalues, singular values, and eigenvectors 60E05 General theory of probability distributions 82B26 Phase transitions (general) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60K25 Queueing theory