Khintchine inequalities, trace monoids and Turán-type problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909634872213504 |
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| author | Santos, Patrick Oliveira Tripathi, Raghavendra Youssef, Pierre |
| author_facet | Santos, Patrick Oliveira Tripathi, Raghavendra Youssef, Pierre |
| contents | 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 Turán-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_02517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Khintchine inequalities, trace monoids and Turán-type problems Santos, Patrick Oliveira Tripathi, Raghavendra Youssef, Pierre Operator Algebras Combinatorics Functional Analysis Probability 46L54, 47A30, 46L89, 46B09, 05C35 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 Turán-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. |
| title | Khintchine inequalities, trace monoids and Turán-type problems |
| topic | Operator Algebras Combinatorics Functional Analysis Probability 46L54, 47A30, 46L89, 46B09, 05C35 |
| url | https://arxiv.org/abs/2506.02517 |