Given a diagonalizable matrix $A \in \mathbb{R}^{n \times n}$ with real eigenvalues, satisfying $1+c_1 \le \rho(A) \le 1+c_2$ $(0<c_1 \le c_2)$, obviously there exists a $v \in \mathbb{R}^{n}$ such that $\|Av\| \ge \|v\|$.
I am interested in the case where $v$ is drawn uniformly at random from some discrete distribution, for instance $\{\pm1\}^{n}$. Are there any known lower bounds on $\Pr[\|Av\| \ge \|v\|]$?
Thank you.