Coexistence of infinite clusters for percolation and Ising model on ℤd
Abstract
For independent bond percolation on ℤd with parameter p, let pcb(d) be the critical probability. We prove that for each d ≥ 9, there is εd>0 such that for each p ∈ (pcb(d), pcb(d)+εd), the complement of the infinite open cluster stochastically dominates a supercritical site percolation on ℤd. This improves the previous results by Grimmett, Holroyd and Kozma 2014, and Bock, Damron, Newman and Sidoravicius 2020. Numerical estimates of pcb(d) and pcs(d) (the site critical probability) suggest that a similar stochastic domination result holds for all d ≥ 4. For the Ising model on ℤd with inverse temperature β, let βc(d) be the critical inverse temperature. We prove that for each d ≥ 8, there is εd>0 such that for each β∈ [0,βc(d)+εd), the + spins under the minus phase stochastically dominate a supercritical site percolation on ℤd. This improves the previous result of Aizenman, Bricmont and Lebowitz 1987. Numerical estimates of βc(d) and pcs(d) suggest that a similar stochastic domination result holds for all d ≥ 5.
Related
- Branching Gaps, Coexistence, and Boundary Influence for the Ising Model on Spherically Symmetric Trees
- Signed FK percolation in Ising spin glasses
- Phase transition for strongly correlated percolation models on supercritical Bernoulli clusters
- No infinite cluster at criticality for long-range percolation with fast-decaying interactions
- Infinite light rays and infinite clusters with infinitely many pivots
BibTeX
@misc{jiang2026coexistence,
title = {{Coexistence of infinite clusters for percolation and Ising model on $\mathbbZ^d$}},
author = {Jianping Jiang and Sike Lang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29806},
eprint = {2609.29806},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29806},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coexistence of infinite clusters for percolation and Ising model on ℤ^d
AU - Jianping Jiang
AU - Sike Lang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29806
UR - https://arxiv.org/abs/2609.29806
AB - For independent bond percolation on ℤ^d with parameter p, let p_c^b(d) be the critical probability. We prove that for each d ≥ 9, there is ε_d>0 such that for each p ∈ (p_c^b(d), p_c^b(d)+ε_d), the complement of the infinite open cluster stochastically dominates a supercritical site percolation on ℤ^d. This improves the previous results by Grimmett, Holroyd and Kozma 2014, and Bock, Damron, Newman and Sidoravicius 2020. Numerical estimates of p_c^b(d) and p_c^s(d) (the site critical probability) suggest that a similar stochastic domination result holds for all d ≥ 4. For the Ising model on ℤ^d with inverse temperature β, let β_c(d) be the critical inverse temperature. We prove that for each d ≥ 8, there is ε_d>0 such that for each β∈ [0,β_c(d)+ε_d), the + spins under the minus phase stochastically dominate a supercritical site percolation on ℤ^d. This improves the previous result of Aizenman, Bricmont and Lebowitz 1987. Numerical estimates of β_c(d) and p_c^s(d) suggest that a similar stochastic domination result holds for all d ≥ 5.
ER -