Asymptotic expansion for transport maps between laws of multimatrix models
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Jekel, David Nikitopoulos, Evangelos A. Parraud, Félix |
| author_facet | Jekel, David Nikitopoulos, Evangelos A. Parraud, Félix |
| contents | We study the large-$N$ behavior of random matrix tuples $Y^N = (Y_1^N,\dots,Y_d^N)$ with joint density proportional to $e^{-N^2 V}$ for some convex function $V$ in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of $1/N^2$ of the trace of noncommutative smooth functions of $Y^N$. We also give an asymptotic expansion for a family of maps $T^N$ that transport the law of a tuple of independent GUE random matrices to the law of $Y^N$ and, as a consequence, show strong convergence for the multimatrix models $Y^N$. Our proof is based on an asymptotic expansion for the heat semigroup associated to the measure, which is expressed in terms of smooth functions of a matrix Brownian motion $(S^{N}_t)_{t \geq 0}$. We introduce spaces of noncommutative smooth functions that unify and generalize the cases of polynomials and single-variable smooth functions and allow the systematic application of asymptotic expansion techniques to multimatrix models with convex interaction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03213 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansion for transport maps between laws of multimatrix models Jekel, David Nikitopoulos, Evangelos A. Parraud, Félix Probability Operator Algebras 46L54, 60B20, 35Q49, 46L52, 60H10, 41A60 We study the large-$N$ behavior of random matrix tuples $Y^N = (Y_1^N,\dots,Y_d^N)$ with joint density proportional to $e^{-N^2 V}$ for some convex function $V$ in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of $1/N^2$ of the trace of noncommutative smooth functions of $Y^N$. We also give an asymptotic expansion for a family of maps $T^N$ that transport the law of a tuple of independent GUE random matrices to the law of $Y^N$ and, as a consequence, show strong convergence for the multimatrix models $Y^N$. Our proof is based on an asymptotic expansion for the heat semigroup associated to the measure, which is expressed in terms of smooth functions of a matrix Brownian motion $(S^{N}_t)_{t \geq 0}$. We introduce spaces of noncommutative smooth functions that unify and generalize the cases of polynomials and single-variable smooth functions and allow the systematic application of asymptotic expansion techniques to multimatrix models with convex interaction. |
| title | Asymptotic expansion for transport maps between laws of multimatrix models |
| topic | Probability Operator Algebras 46L54, 60B20, 35Q49, 46L52, 60H10, 41A60 |
| url | https://arxiv.org/abs/2604.03213 |