Gaussian block random matrix determinant conjecture

Let $m\in\mathbb{N}$ and $a_i\in M_m(\mathbb{Z})$ be given. Define the completely positive map

$$\eta:M_m(\mathbb{C})\to M_m(\mathbb{C}),\qquad b\mapsto \eta(b):=\sum_{i=1}^n a_i b a_i^*.$$

Assume that $\eta$ is rank non-decreasing. For $n$ independent GUE random matrices $X^{(N)}_1,\dots,X^{(N)}_n$, let

$$A_N=\sum_{i=1}^n a_i\otimes X^{(N)}_i$$

be the resulting Gaussian block random matrix of size $mN\times mN$. Gaussian block determinant conjecture. The matrix $A_N$ is invertible for sufficiently large $N$, and

$$\lim_{N\to\infty}\Delta(A_N)=\operatorname{cap}(\eta)^{\frac{1}{2m}}e^{-\frac{1}{2}},$$

both in expectation and almost surely. This conjecture extends the determinantal property known for free-group generators and free semicircular operators to Gaussian block random matrices; the rank non-decreasing hypothesis is the condition under which the capacity expression is expected to govern the limiting Fuglede–Kadison determinant, while the asserted invertibility and convergence remain open in the stated generality.