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 Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of Xz for different complex shift parameters z using the Dyson Brownian Motion.

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This can be explicitly computed in the Ginibre case while for the general i.i.d. case one can infer from the universality of local correlation functions near the edge.49 

34.

The iterative cumulant expansion has been systematically developed in Refs. 62 and 64 extending the iterative gain from so-called un-matched indices.61,65 and exploiting that the leading deterministic terms may cancel in certain situations.66–68 

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Although there is no direct relation between the eigenvalues σ ∈ Spec(X) and the singular values of Hz apart from the trivial fact that z = σ is an eigenvalue if and only if Hz has a zero singular value, we still expect their correlation decay to be similar.

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The matrix entries of X do not have to be identically distributed. Our proof still works with minor modifications if Exij=Exij2=0, E|xij|2 = 1/n and E|nxij|pCp, but for simplicity we consider the i.i.d. case only.

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