Mathematics – Probability
Scientific paper
2010-10-08
Mathematics
Probability
29 pages, 18 figures
Scientific paper
The universality phenomenon asserts that the distribution of the eigenvalues of random matrix with iid zero mean, unit variance entries does not depend on the underlying structure of the random entries. For example, a plot of the eigenvalues of a random sign matrix, where each entry is +1 or -1 with equal probability, looks the same as an analogous plot of the eigenvalues of a random matrix where each entry is complex Gaussian with zero mean and unit variance. In the current paper, we prove a universality result for sparse random $n$ by $n$ matrices where each entry is non-zero with probability $1/n^{1-\alpha}$ where $0 < \alpha \le 1$ is any constant. The sparse universality result proves convergence in probability and has one additional hypothesis that the real and imaginary parts of the entries are independent (this hypothesis is most likely an artifact of the proof). One consequence of the sparse universality principle is that the circular law holds for sparse real random matrices so long as the entries have zero mean and unit variance, which is the most general result for sparse real matrices to date.
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