Let $f(x) = f(x_1, . . . , x_n)$ be a polynomial of degree $d$ and $\text{Var}[f] = 1$. One result by Carbery and Wright shows that for any $t\in\mathbb{R}$ and $ε > 0$, $$ \text{Pr}_{x\sim N^n}[|f(x) − t| ≤ ε] ≤ O(d) ε^{1/d}\label{1}\tag{$\color{red}{\star}$} $$ and there are some simplified proofs (eg. this one).
My questions:
- Is there any related results about the anti-concentration of the eigenvalues of a random matrix?
- For a real symmetric matrix whose entries are normal variables, can we bound the term $$ \text{Pr}[|\lambda_i-t|\leq\epsilon] $$ where $\lambda_i$ is some eigenvalue (or more specifically, the smallest/largest,etc)?
- Apparently the eigenvalue may not be a polynomial of its entries so we can't directly use the above result. Is there any research or literature about this problem?
\eqref{1} Carbery A, Wright J. Distributional and $ L^{q} $ norm inequalities for polynomials over convex bodies in ${\Bbb R}^ n$[J]. Mathematical Research Letters, 2001, 8(3): 233-248, MR1839474, Zbl 0989.26010.