# Anti concentration of $\frac{1}{n}zz^T + \text{diag}(z_1^2,\ldots,z_n^2)$ for sub-gaussian i.i.d $z_1,\ldots,z_n$ and $z:=(z_1,\ldots,z_n)$

Let $$z_1,\ldots,z_n$$ be an i.i.d sample from a sub-gaussian distribution. Define the $$n$$-by-$$n$$ p.s.d matrix $$C_n := \frac{1}{n}zz^T + \text{diag}(z_1^2,\ldots,z_n^2)$$, where $$z:=(z_1,\ldots,z_n)$$.

# Question

What are some anti-concentration inequalities for $$C_n$$?

By anti-concentration, I mean inequality of the form "$$C_n \succeq cI_n$$" w.h.p., or something similar.