Statistical properties of random weak Latin hypercubes

Determine additional statistical statements about random weak Latin hypercubes, including higher-dimensional analogues of the proposed variance and concentration analysis for random permutation sets.

Background

The paper computes the expected periodic L2-discrepancy of uniformly random weak Latin hypercubes and uses this expectation to obtain asymptotically sharp upper bounds for optimal discrepancy when d ≥ 3. It then asks whether further probabilistic information can be obtained.

For d = 2, the authors report a variance formula for a uniformly random permutation and indicate that it could yield a concentration result via Chebyshev’s inequality, but they have not verified the formula in detail. They explicitly state that they lack a corresponding formula or approach for higher-dimensional weak Latin hypercubes.

References

Which other statistical statements can be made about random (weak) Latin hypercubes? For $d=2$, it seems possible to get a concentration result via Chebyshev's inequality using $$ Var (\sigma)2 = \frac{(N-3)(N-2)2(N-1)2(N+1)2}{16200N5} \sim \frac{N2}{16200}, $$ where $\sigma$ is a randomly chosen (uniformly) permutation of ${0, 1, \dots, N-1}$. We have checked this formula via computer calculations for $N \leq 13$ but we did not verify it in detail. We have no suggestion for a similar formula for higher dimensional (weak) Latin hypercubes.

— On the $L_2$-discrepancy of Latin hypercubes  (2502.20828 - Nagel, 28 Feb 2025) in Section 6, Conclusion and outlook, item (iii)