# What is the probability $\Pr(\|U-I\|_{\rm op}<\varepsilon)$ of a Haar-random unitary being close to the identity?

If one generates an $$n\times n$$ Haar random unitary $$U$$, then clearly $$\Pr(U=I)=0$$. However, for every $$\epsilon>0$$, the probability $$\Pr(\|U-I\|_{\rm op}<\varepsilon)$$ should be positive. How can this quantity be computed?

• Exatly? Approximately? Upper/lower bounds? Analytically? Numerically? – Norbert Schuch Aug 14 '20 at 11:20
• arxiv.org/abs/1506.07259 could contain some useful information or techniques (they compute said quantity, but for a different distance measure). – Norbert Schuch Aug 14 '20 at 11:27
• Let's say we want an analytical lower-bound? – Calvin Liu Aug 15 '20 at 15:08
• @CalvinLiu I'll leave this as a comment having not carefully worked out the details, but you can think of this probability as computing the ratio of the volume of an $\epsilon$-ball around $I$ to the volume of the unitary group (with respect to the operator norm), equivalently you can think about this as the 1/size of an $\epsilon$-net for $U(n)$. Very roughly, this is going to be ${\rm Pr}(\|U-I\|_\infty \leq \epsilon) \sim (n/\epsilon^2)^{-n^2}$. – 4xion Aug 25 '20 at 15:36
• If you need a rigorous lower bound you should be able to compute this more precisely (look up refs related to volumes of balls in the unitary group and $\epsilon$-nets) – 4xion Aug 25 '20 at 15:38