Non-asymptotic bounds for the average singular value of a complex Gaussian matrix

Let $G_{d}$ be a $d \times d$ matrix with independent standard complex Gaussian entries, let $α_{\mathbb{C}}(d)$ be the expected average singular value of $G_{d}/\sqrt{d}$, and set $Δ_d := α_{\mathbb{C}}(d)-α_{\mathbb{C}}(d+1)$. The statistic $α_{\mathbb{C}}(d)$ admits a variational representation a