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Real eigenvalues of non-Gaussian random matrices and their products

We study the properties of the eigenvalues of real random matrices and their products. It is known that when the matrix elements are Gaussian-distributed independent random variables, the fraction of real eigenvalues tends to unity as the number of matrices in the product increases. Here we present numerical evidence that this phenomenon is robust with respect to the probability distribution of matrix elements, and is therefore a general property that merits detailed investigation. Since the elements of the product matrix are no longer distributed as those of the single matrix nor they remain independent random variables, we study the role of these two factors in detail. We study numerically the properties of the Hadamard (or Schur) product of matrices and also the product of matrices whose entries are independent but have the same marginal distribution as that of normal products of matrices, and find that under repeated multiplication, the probability of all eigenvalues to be real increases in both cases, but saturates to a constant below unity showing that the correlations amongst the matrix elements are responsible for the approach to one. To investigate the role of the non-normal nature of the probability distributions, we present a thorough analytical treatment of the $2 \times 2$ single matrix for several standard distributions. Within the class of smooth distributions with zero mean and finite variance, our results indicate that the Gaussian distribution has the maximum probability of real eigenvalues, but the Cauchy distribution characterised by infinite variance is found to have a larger probability of real eigenvalues than the normal. We also find that for the two-dimensional single matrices, the probability of real eigenvalues lies in the range [5/8,7/8].