Simplicity of singular value spectrum of random matrices and two-point quantitative invertibility
Yi Han
TL;DR
The paper resolves two central questions for i.i.d. subgaussian random matrices: (i) how likely is every singular value to be simple, and (ii) how likely are two simultaneous small least singular values for two shifted matrices A−λ_1 I and A−λ_2 I? By combining a linearization strategy with inverse Littlewood–Offord theory and the novel essential LCD concept, the authors prove that all singular values are simple with probability at least 1−e^{−cn}, and they establish joint two-point bounds that decay like ε^2 (or ε^4 in the complex case) with an additional factor depending on |λ_1−λ_2|. Key tools include no-gaps delocalization, randomized nets built via inversion of randomness, and a Fourier replacement principle to transfer small-ball probabilities to invertibility statements. The results extend to rectangular matrices and complex shifts, yielding strong anticoncentration consequences for linear combinations of eigenvalues and contributing substantial progress on longstanding conjectures about spectral simplicity and two-location invertibility. Overall, the work provides a robust framework for controlling arithmetic structure in random matrices and yields sharp probabilistic bounds with strong implications for stability and spectral uniqueness in high dimensions.
Abstract
Let $A$ be an $n\times n$ random matrix with independent, identically distributed mean 0, variance 1 subgaussian entries. We prove that $$ \mathbb{P}(A\text{ has distinct singular values})\geq 1-e^{-cn} $$ for some $c>0$, confirming a conjecture of Vu. This result is then generalized to singular values of rectangular random matrices with i.i.d. entries. We also prove that for two fixed real numbers $λ_1,λ_2$ with a sufficient lower bound on $|λ_1-λ_2|$, we have a joint singular value small ball estimate for any $ε>0$ $$ \mathbb{P}(σ_{min}(A-λ_1I_n)\leqεn^{-1/2},σ_{min}(A-λ_2I_n)\leqεn^{-1/2})\leq Cε^2+e^{-cn}, $$ where $σ_{min}(A)$ is the minimal singular value of a square matrix $A$ and $I_n$ is the identity matrix. For much smaller $|λ_1-λ_2|$ we derive a similar estimate with $C$ replaced by $C\sqrt{n}/|λ_1-λ_2|$. This generalizes the one-point estimate of Rudelson and Vershynin, which proves $\mathbb{P}(σ_{min}(A)\leq εn^{-1/2})\leq Cε+e^{-cn}$. Analogous two-point bounds are proven when $A$ has i.i.d. real and complex parts, with $ε^4$ in place of $ε^2$ on the right hand side of the estimate and for any complex numbers $λ_1,λ_2$. These two point estimates can be used to derive strong anticoncentration bounds for an arbitrary linear combination of two eigenvalues of $A$.