Circular law, extreme singular values and potential theory |
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Authors: | Guangming Pan |
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Institution: | a Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore b Department of Statistics and Applied Probability, National University of Singapore, Singapore 117546, Singapore |
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Abstract: | Consider the empirical spectral distribution of complex random n×n matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements. |
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Keywords: | primary 15A52 60F15 secondary 31A15 |
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