Abstract We study bond and site Bernoulli percolation models on Zᵈ for d 3 with parameter p, in both the oriented and non-oriented versions. The main macroscopic quantity of interest is the probability of long-range order, and the existence of a non-trivial threshold is well established. Precise numerical results for the threshold values are available in the literature, but mathematically rigorous bounds are mostly restricted to two-dimensional lattices. Utilizing dynamical coupling techniques, we introduce a comprehensive set of new rigorous upper bounds that corroborate existing numerical values.
Gomes et al. (2026) studied this question.