Removal of Catalan multiplicities

Derive a general spectral-counting argument for free Laplacians satisfying CD(c,∞) that removes the Catalan-number loss and yields the sharper multiplicity behavior found in the semicircular case.

Background

The spectral descent argument gives an exponential upper bound for the spectral counting function, but the general estimate involves Catalan numbers, whereas the finite semicircular model has exact multiplicities nk. The paper attributes the discrepancy to a rigid conditional-expectation slicing argument available in the semicircular case and explicitly states that it does not know how to generalize that method.

References

We have not, however, been able to push the argument far enough to remove the Catalan numbers. The semicircular case has an additional very rigid structure which can be explained by ``slicing'' Dabrowski's almost-commutation formula using the conditional expectation \tau\otimes\mathrm{id}. This leads to the exact multiplicities nk, instead of the Catalan multiplicities appearing in our general argument. We do not know for now how to adapt this method in the general case.

— Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound  (2610.22109 - Diez, 15 Oct 2026) in Section 1, subsection “Positive free curvature and the spectral Bochner argument”

Let $X=(X_1,\ldots,X_n)$ satisfy $CD(c,\infty)$ for some $c>0$ and denote its free Laplacian by $\Delta=\partial*\bar\partial$. Then we have

\lambda_k(\Delta) \geq \lambda_k(c\mathcal L_n), \ k\geq0.

— Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound  (2610.22109 - Diez, 15 Oct 2026) in Conjecture “Free version of E. Milman’s comparison,” Section 3, subsection “Further questions on spectral comparison”

What remains open, even in dimension one, is whether the same spectral comparison follows directly from the curvature condition, without requiring a free Caffarelli contraction theorem.

— Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound  (2610.22109 - Diez, 15 Oct 2026) in Final paragraph of Section 3, subsection “Further questions on spectral comparison”