We address the following question: given a collection $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$ of independent $d \times d$ random matrices drawn from a common distribution $\mathbb{P}$, what is the probability that the centralizer of $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$ is trivial? We provide lower bounds on this probability in terms of the sample size $N$ and the dimension $d$ for several families of random matrices which arise from the discretization of linear Schrödinger operators with random potentials. When combined with recent work on machine learning theory, our results provide guarantees on the generalization ability of transformer-based neural networks for in-context learning of Schrödinger equations.

翻译：我们探讨以下问题：给定从共同分布 $\mathbb{P}$ 中抽取的 $d \times d$ 独立随机矩阵集合 $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$，其中心化子为平凡的概率是多少？针对源自具有随机势的线性薛定谔算子离散化的若干随机矩阵族，我们以样本量 $N$ 和维度 $d$ 为变量给出了该概率的下界。结合近期机器学习理论的研究成果，我们的结果为基于Transformer的神经网络在薛定谔方程上下文学习中的泛化能力提供了理论保证。