Handling tensoring with deterministic matrices in the asymptotic expansion and transport framework

Develop an extension of the asymptotic expansion and transport map framework introduced in this work to accommodate tensoring random matrix tuples with deterministic matrices, potentially by incorporating operator-space-like structures into the spaces of noncommutative smooth functions.

Background

The paper constructs asymptotic expansions for expectations and transport maps using new noncommutative smooth function spaces tailored to convex multimatrix models. In light of recent strong convergence results, it is natural to ask whether these techniques can be upgraded to incorporate tensoring with deterministic matrices.

The authors note that achieving this may require adding operator-space-like structure to their function spaces, and they explicitly defer this direction to future work.

References

Finally, it is natural in light of the strong convergence literature to ask to what extent the theorems and methods in our work can be upgraded to handle tensoring with deterministic matrices. This might require incorporating operator-space like structure into our spaces of smooth functions. We leave this problem as an interesting direction for future research.

— Asymptotic expansion for transport maps between laws of multimatrix models  (2604.03213 - Jekel et al., 3 Apr 2026) in Introduction (end of Section 1)