Eigenvalue Bounds for Random Matrices via Zerofreeness
Sidhanth Mohanty, Amit Rajaraman
TL;DR
The paper introduces a Jensen's formula–based, zerofreeness approach to bound the spectral radius $\rho(\boldsymbol{M})$ of random matrices in a non-asymptotic setting. By relating outlier eigenvalues to the averaged determinant $\mathbb{E}|\det(I - z\boldsymbol{M})|^2$ and connecting $\rho(\boldsymbol{M})$ to the nonbacktracking matrix $B_{\boldsymbol{M}}$, the authors derive constant-probability eigenvalue bounds for three ensembles: non-asymptotic Girko matrices, non-asymptotic Wigner matrices, and the configuration-model random $d$-regular graphs. They establish a general moment-bounds framework (assumption on mixed moments) that yields bounds on $\mathbb{E}\rho(\boldsymbol{M})^2$ and tail bounds for outliers, and they verify these conditions for the random $d$-regular ensemble via detailed combinatorial and analytic arguments including Laplace’s method. The results recover classical bounds for Wigner matrices and Friedman's constant-probability bounds for random regular graphs up to universal constants, offering a simple, non-asymptotic route to spectral radius control in sparse regimes with potential broad applicability.
Abstract
We introduce a new technique to prove bounds for the spectral radius of a random matrix, based on using Jensen's formula to establish the zerofreeness of the associated characteristic polynomial in a region of the complex plane. Our techniques are entirely non-asymptotic, and we instantiate it in three settings: (i) The spectral radius of non-asymptotic Girko matrices -- these are asymmetric matrices $\mathbf{M} \in \mathbb{C}^{n \times n}$ whose entries are independent and satisfy $\mathbb{E} \mathbf{M}_{ij} = 0$ and $\mathbb{E} |\mathbf{M}_{ij}^2| \le \frac{1}{n}$. (ii) The spectral radius of non-asymptotic Wigner matrices -- these are symmetric matrices $\mathbf{M} \in \mathbb{C}^{n \times n}$ whose entries above the diagonal are independent and satisfy $\mathbb{E} \mathbf{M}_{ij} = 0$, $\mathbb{E} |\mathbf{M}_{ij}^2| \le \frac{1}{n}$, and $\mathbb{E} |\mathbf{M}_{ij}^4| \le \frac{1}{n}$. (iii) The second eigenvalue of the adjacency matrix of a random $d$-regular graph on $n$ vertices, as drawn from the configuration model. In all three settings, we obtain constant-probability eigenvalue bounds that are tight up to a constant. Applied to specific random matrix ensembles, we recover classic bounds for Wigner matrices, as well as results of Bordenave--Chafaï--García-Zelada, Bordenave--Lelarge--Massoulié, and Friedman, up to constants.