2026/07/31 by Kosuke Suzuki
Mathematics · Computer Science · #math.NA #cs.NA #msc:11K36 #msc:11K45 #msc:52C17
arxiv created 2026/07/31 · arxiv updated 2026/08/03
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 Θℙ(N1/d(log N)1/d) for Monte Carlo sampling, Θℙ(N1/(d+1)) for jittered sampling, and, for d≥ 2, Θℙ(N1/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 Oℙ(N-3/(2d)), and its mesh ratio is therefore Ωℙ(N1/(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≥ 2, the mesh ratio is Oℙ(log N), whereas it is Θℙ(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.