Paper detail

On the least singular value of random symmetric matrices

Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by $n^γ$ for some $γ>0$. Consider a randomly perturbed matrix $M_n=F_n+X_n$, where $X_n$ is a random symmetric matrix whose upper diagonal entries $x_{ij}$ are iid copies of a random variable $ξ$. Under a very general assumption on $ξ$, we show that for any $B>0$ there exists $A>0$ such that $P(σ_n(M_n)\le n^{-A})\le n^{-B}$. The proof uses an inverse-type result concerning concentration of quadratic forms, which is of interest of its own.