多矩阵模型之间传输映射的渐近展开
Asymptotic expansion for transport maps between laws of multimatrix models
AI总结：
本文研究了大N行为，推导了非交换光滑函数的迹的渐近展开，以及传输映射的渐近展开，展示了多矩阵模型的强收敛性。
摘要：
我们研究了随机矩阵元组Y^N=(Y_1^N,…,Y_d^N)在大-N行为下的性质，其联合密度与e^{-N^2 V}成正比，其中V是非交换变量中的凸函数，且其二阶导数满足一定限制。我们给出了非交换光滑函数的迹的渐近展开，以及传输映射T^N的渐近展开，该映射将独立GUE随机矩阵元组的分布传输到Y^N的分布，并由此展示了多矩阵模型Y^N的强收敛性。我们的证明基于与测度相关的热半群的渐近展开，该展开以矩阵布朗运动(S^{N}_t)_{t≥0}的光滑函数形式表达。我们引入了非交换光滑函数的空间，统一并推广了多项式和单变量光滑函数的情况，允许系统地应用渐近展开技术到具有凸相互作用的多矩阵模型。
英文摘要：
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.
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