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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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