2026-08-17: Dimitri Shlyakhtenko: A free analog of Bobkov’s isoperimetry inequality
(UCLA)
We prove a one-variable functional inequality which is the free probability analog of Bobkov’s isoperimetry inequality. The inequality involves the L1 norm of the difference quotient of a function f and can be viewed as a non-local isoperimetric inequality. We also prove related inequalities for subsets of an interval as well as subsets of roots of Hermite polynomials. This paper is also an experiment in AI-based exploration of free analogs of classical probability statements.
2026-08-24: Enli Chen: Strongly convergent matrix models for q-Gaussian families
(TU Delft)
Let (s_1^{(q)}, \dots, s_r^{(q)}) be a finite q-Gaussian family. We construct explicit finite-dimensional random matrix models that converge strongly to this family for |q| < \sqrt{2} – 1 . Thus, for every noncommutative polynomial evaluated at the random matrix models, both its normalized trace and its operator norm
converge to the corresponding quantities for the q-Gaussian family in probability. The construction combines two ingredients. First, we consider normalized sums of graph-product semicircular variables indexed by Erdos-Renyi random graphs with an additional Clifford twist when q < 0. We prove quantitative operator-norm estimates for the resulting approximate q-Toeplitz relations, as well as an asymptotically sharp degree-one Khintchine inequality. In particular, the normalized
semicircular sum has limiting norm \frac{2}{\sqrt{1-q}} , which is the norm of the standard q-Gaussian variable. Using ultraproduct methods, we upgrade these estimates to complete multivariable strong convergence. Second, we use a refined tensor-GUE strong-convergence theorem to transfer these operator models to finite-dimensional random matrices. A key feature is uniformity for bounded-degree polynomials with matrix coefficients whose dimensions may grow beyond the size of the random matrices. As operator-algebraic consequences, in the above range of q, the associated q-Gaussian C^*-algebras are MF and, when there are at least two generators, their Brown-Douglas-Fillmore extension semigroups are not groups. This talk is based on recent joint work with Martijn Caspers.