Large deviations of the spectral radius of iid subgaussian random matrices

We study large deviations of the spectral radius of $X_n=n^{-1/2}(ξ_{ij})_{i,j=1}^n$, where the entries are iid, centered, variance-one, and subgaussian, with zero pseudovariance in the complex case. For every fixed $r>1$, we prove $$ \liminf_{n\to\infty}\frac1n\log\mathbb{P}\{ρ(X_n)>r\} \ge -\f