Characterize the commutativity measured by the covariance parameter

Characterize in what sense the parameter v(X) measures the commutativity or non-commutativity of the coefficient matrices A_1, ..., A_n in the Gaussian series X = sum_{i=1}^n g_i A_i, given that this interpretation is unclear from the definition of v(X).

Background

For the Gaussian series random matrix X = sum_{i=1}n g_i A_i, earlier work uses the covariance parameter v(X) to refine the non-commutative Khintchine inequality and to quantify when the matrix behaves as though its summands were freely independent. The paper observes that commutativity should govern whether logarithmic factors occur in norm bounds.

The unresolved issue is that the definition of v(X) does not make clear how, or in what precise mathematical sense, it measures commutativity. Resolving this would clarify the structural meaning of the parameter and explain its role in intrinsic-freeness estimates.

References

In the theory of , this is measured by the parameter $v(X)$; however, it is unclear from its definition in what sense this parameter actually measures commutativity.

— Matrix Concentration and Equivalent Operators on Fock Spaces  (2610.01982 - Bandeira et al., 1 Oct 2026) in Section 1, Introduction, paragraph beginning “However, a few aspects of this theory remain lacking”