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This paper proposes a new algorithm, named Householder Dice (HD), forsimulating dynamics on dense random matrix ensembles with translation-invariantproperties. Examples include the Gaussian ensemble, the Haar-distributed randomorthogonal ensemble, and their complex-valued counterparts. A "direct" approachto the simulation, where one first generates a dense $n \times n$ matrix fromthe 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 theprinciple of deferred decisions: rather than fixing the entire random matrix inadvance, it lets the randomness unfold with the dynamics. At the heart of thismatrix-free algorithm is an adaptive and recursive construction of (random)Householder reflectors. These orthogonal transformations exploit the groupsymmetry of the matrix ensembles, while simultaneously maintaining thestatistical correlations induced by the dynamics. The memory and computationcosts 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 isnearly always the case in practice, the new algorithm leads to significantreductions in runtime and memory footprint. Numerical results demonstrate thepromise of the HD algorithm as a new computational tool in the study ofhigh-dimensional random systems.

Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Gaussian and Random Orthogonal Ensembles