Strong matrix models for q-Gaussian systems beyond the known range

Construct strong random-matrix models for \(q\)-Gaussian systems with any number of variables in the remaining parameter range \(q\ge\sqrt{2}-1\), thereby resolving the open case beyond the currently established range \(q<\sqrt{2}-1\).

Background

The paper places its Kikuchi-matrix results in the broader problem of realizing prescribed noncommutative systems as limits of random matrices. Weak matrix models for qq-Gaussian systems are known, and strong models have been constructed for restricted ranges of qq.

The cited prior result constructs strong matrix models for arbitrary numbers of variables when q<2−1q<\sqrt2-1. The paper explicitly records that the complementary range remains unresolved. This is a contextual open problem rather than a problem solved by the Kikuchi-matrix results themselves.

References

Very recently, strong matrix models were constructed in the range $q<\sqrt 2-1$, for any $k$; the case of larger $q$ remains open.

— Convergence of Kikuchi matrices to $Γ$-independent and $q$-Gaussian limits  (2610.02008 - Bandeira et al., 1 Oct 2026) in Section 1, subsection “Connections and broader context,” paragraph “Matrix models for noncommutative independence” (page number unavailable)