Precise asymptotics for the spectral radius of a large random matrix

Giorgio Cipolloni; László Erdős; Yuanyuan Xu

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arXiv · 2210.15643

Precise asymptotics for the spectral radius of a large random matrix

Giorgio Cipolloni·László Erdős·Yuanyuan Xu

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Abstract

We consider the spectral radius of a large random matrix $X$ with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of $X$ in [29]. To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of $X-z$ for different complex shift parameters $z$ using the Dyson Brownian Motion.

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Giorgio Cipolloni, László Erdős, Yuanyuan Xu. 2022-10-27. Precise asymptotics for the spectral radius of a large random matrix. https://arxiv.org/abs/2210.15643

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