Extreme eigenvalues of Log-concave Ensemble
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In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices XX^* with isotropic log-concave X-columns. A main example is the covariance estimator of the uniform measure on isotropic convex body. Non-asymptotic estimates and first order asymptotic limits for the extreme eigenvalues have been obtained in the literature. In this paper, with the recent advancements on log-concave measures <cit.>, we take a step further to locate the eigenvalues with a nearly optimal precision, namely, the spectral rigidity of this ensemble is derived. Based on the spectral rigidity and an additional “unconditional" assumption, we further derive the Tracy-Widom law for the extreme eigenvalues of XX^*, and the Gaussian law for the extreme eigenvalues in case strong spikes are present.
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