Power law deformation of Wishart-Laguerre ensembles of random matrices
Sep, 2008Citations per year
Abstract: (IOP)
We introduce a one-parameter deformation of the Wishart–Laguerre or chiral ensembles of positive definite random matrices with Dyson index β = 1,2 and 4. Our generalized model has a fat-tailed distribution while preserving the invariance under orthogonal, unitary or symplectic transformations. The spectral properties are derived analytically for finite matrix sizeN×Mfor all three values of β, in terms of the orthogonal polynomials of the standard Wishart–Laguerre ensembles. For largeNin a certain double-scaling limit we obtain a generalized Marčenko–Pastur distribution on the macroscopic scale, and a generalized Bessel law at the hard edge which is shown to be universal. Both macroscopic and microscopic correlations exhibit power law tails, where the microscopic limit depends on β and the differenceM−N. In the limit where our parameter governing the power law goes to infinity we recover the correlations of the Wishart–Laguerre ensembles. To illustrate these findings, the generalized Marčenko–Pastur distribution is shown to be in very good agreement with empirical data from financial covariance matrices.Note:
- 28 pages, 9 figures; v2 published version with typos corrected
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