Let $A \in \mathbb{C}^{n \times n}$ be some (fixed) matrix with eigenvalues $\lambda_{1},\ldots,\lambda_{n}$. Let $E$ be some random, small-normed, perturbation such that $\tilde{A} = A+E$ has eigenvalues $\tilde{\lambda}_1,\ldots,\tilde{\lambda}_{n}$.

Denote $\Delta = \min_{i \neq j}|\tilde{\lambda}_i - \tilde{\lambda}_j|$. Nguyen, Tao and Vu proved that, roughly, if $A$ is Hermitian and $E$ is a random Wigner matrix then for every fixed $a$ there exists $b = b(\|E\|)$ such that $\Delta \ge n^{-b}$ with probability at least $1-n^{-a}$.

From what I understand, no similar result concerning non-Hermitian matrices is known. Note that the specific distribution $E$ is drawn from is not important to me (as long as it is "standard").

I am looking for a slightly weaker result concerning the size of the Jordan blocks of the perturbed matrix. Namely, let $B$ be the size of the largest Jordan block of a Jordan decomposition of $A$. What is w.h.p. the size of the largest Jordan block of $\tilde{A}$?