Large deviations of the spectral radius of iid subgaussian random matrices

We study large deviations of the spectral radius of $X_n=n^{-1\mathrm{/}2}({\xi }_{ij}{)}_{i,j=1}^n$, where the entries are iid, centered, variance-one, and subgaussian, with zero pseudovariance in the complex case. For every fixed $r>1$, we prove $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{1}{n}\log \mathbb{P}\{\rho (X_n)>r\}\ge -\frac{\beta }{2}(r^2-1-2\log r),$$ where $\beta =1$ for real entries and $\beta =2$ for complex entries. We prove a matching upper bound for real symmetric laws with Gaussian-dominated even moments and for complex laws satisfying the sharp Gaussian Laplace-transform bound. These classes include discrete distributions. In the complex sharp class, we also identify the exponential rate for an eigenvalue to enter a fixed disk outside the unit disk. We then turn to lower deviations and establish a quadratic-speed bound. Under the additional assumption that the entry law $\mu$ has a bounded density, we prove $$\mathbb{P}\{\rho (X_n)\le r\}\le e^{-c_{\mu ,r}{n}^2}$$ for every fixed $0<r<1$ and all sufficiently large $n$, where $c_{\mu ,r}>0$. For the matching upper-tail classes with a bounded density, these bounds give a full speed-$n$ large deviation principle, with the displayed upper-tail rate for $r\ge 1$ and infinite rate for $r<1$. The upper-tail proof uses a change of measure that preserves the entry support and creates an outlying eigenvalue. The lower-tail proof develops a weighted comparison for adaptive orthonormal observations and applies it to Arnoldi residuals.