Householder Dice：一种模拟高斯和随机正交系综动力学的无矩阵算法
Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Gaussian and Random Orthogonal Ensembles
摘要：
本文提出了一种新的算法，命名为Householder Dice（HD），用于模拟稠密随机矩阵系综上的动力学具有不变性的性质。例子包括高斯系综，哈尔分布随机正交系综，以及它们的复值对应物。模拟的“直接”方法，其中首先从系综生成密集的$n \times n$矩阵，在空间和时间上至少需要$\mathcal{O}(n^2)$资源。HD算法通过使用延迟决策的原理克服了这个$\mathcal{O}(n^2)$瓶颈：它不是预先固定整个随机矩阵，而是让随机性随动态展开。这种无矩阵算法的核心是（随机）Householder反射器的自适应和递归构造。这些正交变换利用矩阵系综的群对称性，同时保持由动力学引起的统计相关性。HD算法的存储器和计算成本分别为$\mathcal{O}(nT)$和$\mathcal{O}(nT^2)$，其中$T$为迭代次数。当$T \ll n$，这是几乎总是在实践中的情况下，新的算法导致显着减少运行时和内存占用。数值结果表明，HD算法作为一种新的计算工具，在高维随机系统的研究的承诺。
英文摘要：
This paper proposes a new algorithm, named Householder Dice (HD), for simulating dynamics on dense random matrix ensembles with translation-invariant properties. Examples include the Gaussian ensemble, the Haar-distributed random orthogonal ensemble, and their complex-valued counterparts. A "direct" approach to the simulation, where one first generates a dense $n \times n$ matrix from the ensemble, requires at least $\mathcal{O}(n^2)$ resource in space and time. The HD algorithm overcomes this $\mathcal{O}(n^2)$ bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are $\mathcal{O}(nT)$ and $\mathcal{O}(nT^2)$, respectively, with $T$ being the number of iterations. When $T \ll n$, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint. Numerical results demonstrate the promise of the HD algorithm as a new computational tool in the study of high-dimensional random systems.
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