Precise asymptotics for the spectral radius of a large random matrix

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.