Variational Formulas for the Spectrum of Block Wishart Matrices

We analyze the asymptotics of a block-Wishart random matrix ensemble of the type ${\mathit{W}}_k=({\mathit{X}}^{\ast }\otimes {\mathit{I}}_k)\mathit{T}(\mathit{X}\otimes {\mathit{I}}_k)$ for $\mathit{X}\in {\mathbb{C}}^{n\times p}$ with i.i.d. rows satisfying a suitable concentration-of-measure property, and $\mathit{T}:=\mathbf{\text{Diag}}({\mathit{T}}_i{)}_{i\in [n]}$ a block diagonal matrix with self-adjoint blocks ${\mathit{T}}_i\in {\mathbb{C}}^{k\times k}$, under the proportional asymptotics $n\mathrm{/}p\to \alpha$ 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 ${\mathit{W}}_k$: the left (equivalently right) edge of its support, and its logarithmic potential.