On a Generalisation of the Marcenko-Pastur Problem
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Publication:6349151
DOI10.1103/PHYSREVE.102.062117arXiv2009.07113MaRDI QIDQ6349151
Marc Potters, Jean-Philippe Bouchaud
Publication date: 15 September 2020
Abstract: We study the spectrum of generalized Wishart matrices, defined as $mathbf{F}=( X Y^ op + Y X^ op)/2T$, where $X$ and $Y$ are $N imes T$ matrices with zero mean, unit variance IID entries and such that $mathbb{E}[X_{it} Y_{jt}]=c delta_{i,j}$. The limit $c=1$ corresponds to the Marcenko-Pastur problem. For a general $c$, we show that the Stietjes transform of $mathbf{F}$ is the solution of a cubic equation. In the limit $c=0$, $T gg N$ the density of eigenvalues converges to the Wigner semi-circle.
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