
Journal of Lie Theory 34 (2024), No. 1, 137169 Copyright Heldermann Verlag 2024 Random εCover on Compact Riemannian Symmetric Space Somnath Chakraborty Fakultät für Mathematik, RuhrUniversität Bochum, Bochum, Germany somnath.chakraborty@rub.de [Abstractpdf] A randomized scheme that succeeds with probability $12\delta$ (for any $\delta>0$) has been devised to construct (1) an equidistributed $\epsilon$cover, and (2) an approximate $(\lambda_r,2)$design  in a compact Riemannian symmetric space $\mathbb M$ of dimension $d_{\mathbb M}$  using $n(\epsilon,\delta)$many Haarrandom isometries of $\mathbb M$, where $$ n(\epsilon,\delta):={\mathcal O}_{\mathbb M} [d_{\mathbb M} (\ln (1/\epsilon) + \log_2 (1/\delta) ) ]\,, $$ and $\lambda_r=\mathcal O_{\mathbb M} (\epsilon^{1\frac{d_{\mathbb M}}2})$ is the $r$th smallest eigenvalue of the LaplaceBeltrami operator on $\mathbb M$. The $\epsilon$cover soproduced can be used to compute the integral of 1Lipschitz functions within additive $\tilde {\mathcal O}_{\mathbb M}(\epsilon)$error, as well as in comparing persistence homology computed from data cloud to that of a hypothetical data cloud sampled from the uniform measure. Keywords: Symmetric space, epsiloncover, (lambda,2)design, equidistributed cover, random isometries, Wasserstein distance, irreducible representations, Casimir operator, LaplaceBeltrami operator, Schrier graph, expander, spectral gap, Markov chain. MSC: 43A85, 53C30, 68W20. [ Fulltextpdf (263 KB)] for subscribers only. 