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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”) \(S = M^{-1}(X - \bar X)(X -\bar X)^T\), where \(X\) are rectangular \(N \times M\) sample matrices whose columns are observation vectors distributed as \(N(0,\Sigma)\), and \(\bar X\) are sample mean vectors. It is assumed that \(N, M \rightarrow\infty\) and \(M/N \rightarrow\gamma^2 > 1\). 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 \((1 + 1/\gamma)^2\). 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)]
\[ \mathbb P(aM^{2/3}(\lambda_1- (1 + 1/\gamma)^2< x) \rightarrow F(x),\tag{1} \] 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 \(r \leq N\) and the eigenvalues \(l_1 \geq l_2 \geq\dots l_N\) of \(\Sigma\) be such that \(l_{r+1} = l_{r+2} = \dots = l_N.\)
Theorem 1 states that
(i) If \(k\leq r\), \(l_1=l_2=\dots=l_k=1+1/\gamma\) and all \(\lambda_j\), \(j=k+1,k+2\dots,r\), are located on a compact within \((0,1 + 1/\gamma)\), then the largest eigenvalue \(\lambda_1\) of \(S\) has the same limiting distribution (1);
(ii) If for some \(k\leq r\), the eigenvalues \(l_1=l_2=\dots=l_k\) are located on a compact within \((1 + 1/\gamma,\infty)\) and all eigenvalues \(l_j\), \(j=k+1,k+2\dots,r\), are on a compact within \((0,l_1)\), then \[ \mathbb P(aM^{1/2}(\lambda_1-c)\leq x)\to G_k(x), \] where \(a > 0\) and \(c < l_1\) depend on \(\gamma\) and \(l_1\), and \(G_k(x)\) is the distribution function for the largest eigenvalue of \(k\times k\) 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:
15B52 Random matrices (algebraic aspects)
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
62E20 Asymptotic distribution theory in statistics
15A18 Eigenvalues, singular values, and eigenvectors
60E05 Probability distributions: general theory
82B26 Phase transitions (general) in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K25 Queueing theory (aspects of probability theory)
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