CD membership of finite q-Gaussian tuples

Determine whether finite q-Gaussian tuples belong to the free curvature-dimension class CD(c,∞) for a substantial range of q.

Background

The paper distinguishes the ordinary free Laplacian from the q-Gaussian number operator. Although the q-Gaussian number operator has compact resolvent, this does not settle compactness for the ordinary free-difference-quotient Laplacian associated with q-Gaussian generators. The authors explicitly identify as unknown whether such tuples satisfy their curvature-dimension condition over a broad range of deformation parameters.

References

There is also a distinction that has to be made between our setting and the $q$--Gaussian case equipped with its $q$--Gaussian number operator (we do not know whether such tuples generally belong to our $CD$ class for a large range of $q$).

— Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound  (2610.22109 - Diez, 15 Oct 2026) in Section 1, subsection “The classical compactness problem and its free counterpart”