Upper bounds for critical probabilities in Bernoulli percolation models
Abstract
Abstract We study bond and site Bernoulli percolation models on $\mathbb{Z}^d$ for $d \geq 3$ 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.
Fields of Study & Subjects
Stochastic processes and statistical mechanics
Markov Chains and Monte Carlo Methods
Theoretical and Computational Physics
Publication details
- Journal / source
- Journal of Applied Probability
- Volume
- 63
- Issue
- 1
- Pages
- 61–72
- Publication date
- 23 January 2026
- Language
- EN