Theorem \[thm:circularWigner\] is identical to the proof given in Section \[sec:linear\].

The proof of Theorem \[thm:circularWigner\] follows the same arguments outlined in Section \[sec:complete\] (taking ${\mathbf}{A}_N = 0$). Indeed, the proof in Section \[sec:complete\] requires three key inputs: Lemma \[lemma:concentrate\], Theorem \[approxthm\], and Theorem \[thm:least-sing-value\]. Thus, in order to complete the proof, we will need versions of these results for the matrix ${\mathbf}{Z}_N$ defined above.

We first observe that both Lemma \[lemma:concentrate\] and Theorem \[approxthm\] hold for the matrix ${\mathbf}{Z}_N$ defined above. Indeed, the matrices ${\mathbf}{Y}_{N,1}$ and ${\mathbf}{Y}_{N,2}$ (as well as their truncated counterparts) are elliptic random matrices. In fact, the proofs of Lemma \[lemma:concentrate\] and Theorem \[approxthm\] do not require any conditions on the correlations $\rho_k$. Therefore, it only remains to prove the following analogue of Theorem \[thm:least-sing-value\].

\[thm:least-sing-valueWigner\] Under the assumptions of Theorem \[thm:prodWigner\], there exists $A >0$ such that for almost every $z \in \mathbb{C}$, almost surely $$\lim_{N \to \infty} {\ensuremath{\mathbf{1}_{\{\sigma_{2N}( {\mathbf}{Z}_N - z{\mathbf}{I}) \leq N^{-A}\}}}} = 0.$$

In order to prove Theorem \[thm:least-sing-valueWigner\], we will need the following result due to Nguyen [@NgLSV]. The version stated in [@NgLSV] requires that the deterministic matrices ${\mathbf}{F}_{N,k}$ be real symmetric. However, the proof given in [@NgLSV] only requires that 