标题： Khinchin不等式，并迹幺半群和Turán型问题 显示英文标题

标题： Khintchine inequalities, trace monoids and Turán-type problems

摘要： 我们证明了关于自由变量和张量独立的半圆变量混合体的标量值和算子值的Khintchine不等式，这些不等式在经典和自由的Khintchine型不等式之间进行插值。 具体来说，我们通过与由图 $G$确定的迹单体的Cayley图的谱半径相关联的方式，刻画了 $G$-独立半圆变量和的范数。 我们的方法依赖于迹单体中的闭路径与这些算子和的范数之间的精确对应关系。 这种对应关系揭示了非交换概率、组合群论和极值图论之间的联系。 特别是，我们提出了受经典交换约束下最大范数增长的Turán型极值问题，并确定了极值构型。 我们希望这里开发的方法和联系将在研究受组合对称性约束的非交换结构时有用。 显示英文摘要

摘要： 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.