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arXiv · 2602.04614

Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals

Benoît Collins·Manasa Nagatsu
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Abstract

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.

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Benoît Collins, Manasa Nagatsu. 2026-02-04. Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals. https://arxiv.org/abs/2602.04614

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