Khintchine inequalities, trace monoids and Tur\'an-type problems

TL;DR

This paper establishes new Khintchine inequalities for mixtures of free and tensor-independent semicircle variables, linking non-commutative probability with combinatorial graph theory and extremal problems.

Contribution

It introduces a novel characterization of operator norms via spectral radii of Cayley graphs, connecting non-commutative probability with combinatorial and extremal graph theory.

Findings

Derived scalar and operator-valued Khintchine inequalities for semicircle variables.

Linked norm estimates to spectral radii of Cayley graphs of trace monoids.

Formulated Turán-type extremal problems for non-commutative structures.

Abstract

We prove scalar and operator-valued Khintchine inequalities for mixtures of free and tensor-independent semicircle variables, interpolating between classical and free Khintchine-type inequalities. Specifically, we characterize the norm of sums of $G$-independent semicircle variables in terms of the spectral radius of the Cayley graph associated with the trace monoid determined by the graph $G$. Our approach relies on a precise correspondence between closed paths in trace monoids and the norms of such operator sums. This correspondence uncovers connections between non-commutative probability, combinatorial group theory, and extremal graph theory. In particular, we formulate Tur\'an-type extremal problems that govern maximal norm growth under classical commutation constraints, and identify the extremal configurations. We hope that the methods and connections developed here will be useful in the study of non-commutative structures constrained by combinatorial symmetries.