### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 2258-2286 |

Number of pages | 29 |

Journal | Annals of Probability |

Volume | 46 |

Issue number | 4 |

DOIs | |

State | Published - 2018 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Combinatorics
- Digraph
- Heavy tail
- Random matrix
- Spectral radius

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## Cite this

*Annals of Probability*,

*46*(4), 2258-2286. https://doi.org/10.1214/17-AOP1228