Separation properties of scrambled digital nets and related random point sets

Abstract

We study how standard randomization procedures affect the local geometry of quasi-Monte Carlo point sets, as measured by their minimum distance and mesh ratio. Although probabilistic selection within structured lattice families can produce quasi-uniform point sets, randomizing an existing low-discrepancy construction need not preserve quasi-uniformity. We first determine sharp probabilistic orders for Monte Carlo, jittered, and Latin hypercube sampling, whose mesh ratios diverge as positive powers of $N$. The orders are $\Theta_{\mathbb{P}}(N^{1/d}(\log N)^{1/d})$ for Monte Carlo sampling, $\Theta_{\mathbb{P}}(N^{1/(d+1)})$ for jittered sampling, and, for $d\ge 2$, $\Theta_{\mathbb{P}}(N^{1/d}(\log N)^{1/d})$ for Latin hypercube sampling. We also obtain a Weibull limit law for the minimum distance of jittered samples. For full Owen scrambling, every family of fixed-$t$ nets has minimum distance $O_{\mathbb{P}}(N^{-3/(2d)})$, and its mesh ratio is therefore $\Omega_{\mathbb{P}}(N^{1/(2d)})$. Under uniform coincidence and common-prefix conditions, these bounds are sharp up to logarithmic factors. Moreover, a single full Owen scrambling of any $(t,d)$-sequence is almost surely non-quasi-uniform. By contrast, for matrix and linear scrambling of binary digital nets with fixed $t$ in dimension $d\ge 2$, the mesh ratio is $O_{\mathbb{P}}(\log N)$, whereas it is $\Theta_{\mathbb{P}}(\log N)$ in the separate balanced-prefix affine-tail model. The model also yields the exact probabilistic order for one-dimensional binary digital $(0,m,1)$-nets under matrix or linear scrambling. These results demonstrate that the geometric effect of randomization is governed by whether it introduces local independence or shared algebraic randomness.