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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:

  1. Is there any related results about the anti-concentration of the eigenvalues of a random matrix?
  2. 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)?
  3. 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.

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Theorem 2 in

Iain M. Johnstone. Zongming Ma. "Fast approach to the Tracy–Widom law at the edge of GOE and GUE." Ann. Appl. Probab. 22 (5) 1962 - 1988, October 2012. https://doi.org/10.1214/11-AAP819

gives, for the largest eigenvalue $\lambda_1$ of a GOE matrix of size $N$, that there exists $C=C(s_0)$ such that for any real $s\ge s_0$, $$ |P\Big((\lambda_1 - \sqrt{2N+1})/(2^{-1/2}n^{-1/6})\le s \Big) - F_1(s)| \le C N^{-2/3} e^{-s/2} $$ where $F_1(s)$ is the CDF of the Tracy-Widom law for $\beta=1$ (https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution). Small ball probabilities can then be obtained using bounds on the density of $F_1$.

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