Large deviations of the spectral radius of iid subgaussian random matrices

Abstract

We study large deviations of the spectral radius of $X_n=n^{-1/2}(\xi_{ij})_{i,j=1}^n$, where the entries are iid, centered, variance-one, and subgaussian, with zero pseudovariance in the complex case. We write $\rho(X_n)$ for its spectral radius, i.e. the largest modulus among its eigenvalues. For every fixed $r>1$, we prove $$ \liminf_{n\to\infty}\frac1n\log\mathbb{P}\{\rho(X_n)>r\} \ge -\frac\beta2\bigl(r^2-1-2\log r\bigr), $$ 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 $00$. 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 $r\ge1$ 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.