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Tails of Random Matrix Diagonal Elements: The Case of the Wishart Inverse

We analytically compute the large-deviation probability of a diagonal matrix element of two cases of random matrices, namely $β=[\vec H^\dagger\vec H]^{-1}_{11}$ and $γ=[\vec I_N+ρ\vec H^\dagger\vec H]^{-1}_{11}$, where $\vec H$ is a $M\times N$ complex Gaussian matrix with independent entries and $M\geq N$. These diagonal entries are related to the "signal to interference and noise ratio" (SINR) in multi-antenna communications. They depend not only on the eigenvalues but also on the corresponding eigenfunction weights, which we are able to evaluate on average constrained on the value of the SINR. We also show that beyond a lower and upper critical value of $β$, $γ$, the maximum and minimum eigenvalues, respectively, detach from the bulk. Responsible for this detachment is the fact that the corresponding eigenvalue weight becomes macroscopic (i.e. O(1)), and hence exerts a strong repulsion to the eigenvalue.