Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915774185078784 |
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| author | Collins, Benoît Nagatsu, Manasa |
| author_facet | Collins, Benoît Nagatsu, Manasa |
| contents | We consider expectations of the form $E [tr h_1(X_1^N)... tr h_r(X_r^N)]$, where $X_i^N$ are self-adjoint polynomials in various independent classical random matrices and $h_i$ are smooth test function and obtain a large $N$ expansion of these quantities, building on the framework of polynomial approximation and Bernstein-type inequalities recently developed by Chen, Garza-Vargas, Tropp, and van Handel. As applications of the above, we prove the higher-order asymptotic vanishing of cumulants for smooth linear statistics, establish a Central Limit Theorem, and demonstrate the existence of formal asymptotic expansions for the free energy and observables of matrix integrals with smooth potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04614 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals Collins, Benoît Nagatsu, Manasa Probability Operator Algebras 60B20, 15B52, 46L54 We consider expectations of the form $E [tr h_1(X_1^N)... tr h_r(X_r^N)]$, where $X_i^N$ are self-adjoint polynomials in various independent classical random matrices and $h_i$ are smooth test function and obtain a large $N$ expansion of these quantities, building on the framework of polynomial approximation and Bernstein-type inequalities recently developed by Chen, Garza-Vargas, Tropp, and van Handel. As applications of the above, we prove the higher-order asymptotic vanishing of cumulants for smooth linear statistics, establish a Central Limit Theorem, and demonstrate the existence of formal asymptotic expansions for the free energy and observables of matrix integrals with smooth potentials. |
| title | Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals |
| topic | Probability Operator Algebras 60B20, 15B52, 46L54 |
| url | https://arxiv.org/abs/2602.04614 |