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On the spectral radius of a random matrix: an upper bound without fourth moment. (English) Zbl 1393.05130

Summary: Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance. In other words, that there are no outliers in the circular law. In this work, we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.

MSC:

05C20 Directed graphs (digraphs), tournaments
15B52 Random matrices (algebraic aspects)
47A10 Spectrum, resolvent
05C80 Random graphs (graph-theoretic aspects)
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