Let $\mathcal{H}$ be a $d$-dimensional Hilbert space equipped with the Haar measure. Levy's lemma says that, for an $L$ -Lipschitz function $f$ on $\mathcal{H}$, the probability that $f(x)$ for a randomly drawn unit vector $x$ deviates from its expected value $\mathbb{E}[f]$ by more than some $\delta > 0$ is at most exponentially small in $d$. Specifically,
$$\Pr(|f(x) - \mathbb{E}[f]| \geq \delta) \leq 2 \exp\left( -\frac{C d \delta^2}{L^2} \right)$$
for some absolute constant $C > 0$.
Suppose that instead of the Haar measure, we take a $t$-design over the unit vectors. Does Levy's lemma, or some modified form of it, hold for sufficiently small $t$? By "sufficiently small" I am thinking of something like: let $f$ be a polynomial in $x$ with bounded degree $k$, then $t = O(k)$ would be reasonable. A common class of functions are of the form $f(x) = \langle x | A | x \rangle$ for a fixed operator $A$, which have $k = 2$. On the other hand, if $t = \Omega(d)$ is necessary, then this would be too large to be useful (but good to know).
