Variational Formulas for the Spectrum of Block Wishart Matrices

We analyze the asymptotics of a block-Wishart random matrix ensemble of the type ${\boldsymbol W}_k = ({\boldsymbol X}^* \otimes {\boldsymbol I}_k){\boldsymbol T}({\boldsymbol X}\otimes{\boldsymbol I}_k)$ for ${\boldsymbol X} \in\mathbb{C}^{n\times p}$ with i.i.d. rows satisfying a suitable concentration-of-measure property, and ${\boldsymbol T} := \textrm{\bf Diag}({\boldsymbol T}_i)_{i\in[n]}$ a block diagonal matrix with self-adjoint blocks ${\boldsymbol T}_i\in \mathbb{C}^{k\times k}$, under the proportional asymptotics $n/p\to\alpha$ with $k$ fixed. These matrices play a prominent role in the analysis of $k$-index models in high-dimensional statistics. By studying the matrix Stieltjes transform of this random matrix model and its inverse ($K$-transform), we derive variational formulas for two functionals of the asymptotic spectral density of ${\boldsymbol W}_k$: the left (equivalently right) edge of its support, and its logarithmic potential.