Remove the additional fractional-power error term in the intrinsic-freeness bound

Establish whether every Gaussian series random matrix X = sum_{i=1}^n g_i A_i satisfies a bound of the form E||X|| leq ||X_free|| + C v(X)(log d)^{1/2} for an absolute constant C geq 0, without the additional fractional-power error term appearing in the paper’s strengthened intrinsic-freeness inequality.

Background

For a Gaussian series X = sum_{i=1}n g_i A_i, the operator X_free = sum_{i=1}n A_i otimes s_i is the deterministic operator formed from freely independent semicircular variables. Previous work posed whether the expected spectral norm could be bounded by ||X_free|| plus an error of order v(X)(log d){1/2}.

The paper proves only a partial answer: its strengthened intrinsic-freeness inequality contains both a square-root and a cube-root error contribution. Thus, the sharper bound without the cube-root contribution remains unresolved.

References

The authors of posed the question of whether a bound on Gaussian series norms holds of the form $E|X| \leq |\mathcal X_{\mathrm{free}| + C\,v(X)(\log d){1/2}$ for an absolute constant $C \geq 0$, which corresponds to~eq:almostprovesBBvHconj without the last term inside the parentheses.

eq:almostprovesBBvHconj:

E∥X∥≤∥Xfree∥(1+8(v(X)2σ(X)2log⁡d)1/2+8(v(X)2σ(X)2log⁡d)1/3).E\|X\| \leq \|\mathcal X_{\mathrm{free}}\|\left(1 + 8\left(\frac{v(X)^2}{\sigma(X)^2}\log d\right)^{1/2} + 8\left(\frac{v(X)^2}{\sigma(X)^2}\log d\right)^{1/3}\right).

— Matrix Concentration and Equivalent Operators on Fock Spaces  (2610.01982 - Bandeira et al., 1 Oct 2026) in Section 1, subsection “Main results,” paragraph following Theorem 1.5 (Theorem \ref{thm:strong-intrinsic})