Large deviations of the spectral radius of iid subgaussian random matrices
Abstract
We study large deviations of the spectral radius of Xn=n−1/2(ξij)i,j=1n, where the entries are iid, centered, variance-one, and subgaussian, with zero pseudovariance in the complex case. We write ρ(Xn) for its spectral radius, i.e. the largest modulus among its eigenvalues. For every fixed r>1, we prove liminfn→∞1/nlogℙ{ρ(Xn)>r} ≥ −β/2(r2−1−2log r), where β=1 for real entries and β=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 μ has a bounded density, we prove ℙ{ρ(Xn)≤ r}≤ e−cμ,rn2 for every fixed 0<r<1 and all sufficiently large n, where cμ,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≥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.
Related
- Extremal Graphs with Prescribed Parameters for the Spectral Radius of Weighted Adjacency Matrices with Property P∗
- Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials
- Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles
- Spectral properties of Random Matrices
- Adaptive Random Matrices in Gaussian Bandits: Spectral Universality and Selection-Induced Outliers
BibTeX
@misc{han2026large,
title = {{Large deviations of the spectral radius of iid subgaussian random matrices}},
author = {Yi Han},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05498},
eprint = {2610.05498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05498},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large deviations of the spectral radius of iid subgaussian random matrices
AU - Yi Han
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.05498
UR - https://arxiv.org/abs/2610.05498
AB - 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. We write ρ(X_n) for its spectral radius, i.e. the largest modulus among its eigenvalues. For every fixed r>1, we prove liminf_n→∞1/nlogℙ{ρ(X_n)>r} ≥ −β/2(r^2−1−2log r), where β=1 for real entries and β=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 μ has a bounded density, we prove ℙ{ρ(X_n)≤ r}≤ e^−c_μ,rn^2 for every fixed 0<r<1 and all sufficiently large n, where c_μ,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≥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.
ER -