Repeated singular values of a random symmetric matrix and decoupled singular value estimates
Yi Han
TL;DR
The paper addresses the problem of two-point control of the least singular values of random symmetric (Wigner) matrices around two separated spectral locations. It develops a two-pronged approach: (I) a refined inversion-of-randomness framework to certify quasi-random arithmetic structure for vector pairs and (II) a decoupling argument that leverages the local semicircle law and Log-Sobolev constants to separate the two location events. The main contributions include a sharp joint singular-value bound with decoupling, a mesoscopic-extension with exponential-type errors, and corollaries ensuring distinct singular values across a bulk interval with high probability, thereby substantially confirming Vu’s conjecture in a quantitative, high-probability sense for large portions of the spectrum. The techniques combine inverse Littlewood-Offord theory for vector pairs, refined net constructions, a bootstrap amplification of probabilistic bounds, and local-law-based spectral rigidity to yield robust two-point and mesoscopic decoupling results with broad applicability to random symmetric matrices beyond the i.i.d. setting. The results advance understanding of spectral rigidity and extreme-value behavior in random matrices, with potential implications for universality and eigenvalue-gap statistics near the bulk.
Abstract
Let $A_n$ be a random symmetric matrix with Bernoulli $\{\pm 1\}$ entries. For any $κ>0$ and two real numbers $λ_1,λ_2$ with a separation $|λ_1-λ_2|\geq κn^{1/2}$ and both lying in the bulk $[-(2-κ)n^{1/2},(2-κ)n^{1/2}]$, we prove a joint singular value estimate $$ \mathbb{P}(σ_{min}(A_n-λ_i I_n)\leqεn^{-1/2};i=1,2)\leq Cε^2+2e^{-cn}. $$ For general subgaussian distribution and a mesoscopic separation $|λ_1-λ_2|\geq κn^{-1/2+σ},σ>0$ we prove the same estimate with $e^{-cn}$ replaced by an exponential type error. This means that extreme behaviors of the least singular value at two locations can essentially be decoupled all the way down to the exponential scale when the two locations are separated. As a corollary, we prove that all the singular values of $A_n$ in $[κn^{1/2},(2-κ)n^{1/2}]$ are distinct with probability $1-e^{-cn}$, and with high probability the minimal gap between these singular values has order at least $n^{-3/2}$. This justifies, in a strong quantitative form, a conjecture of Vu up to $(1-κ)$-fraction of the spectrum for any $κ>0$.