For every finite graph G=(V,E), every edge-parameter vector (p_e)_e∈ E∈[0,1]^E, every q∈[0,1], and every pair of distinct edges e,f∈ E, the random-cluster measure phi_G,mathbf p,q satisfies phi_G,mathbf p,q(e is open,f is open)≤ phi_G,mathbf p,q(e is open),phi_G,mathbf p,q(f is open). For q>0, this measure assigns to each configuration omega∈0,1^E probability proportional to q^k(omega)∏_e∈ Ep_e^omega(e)(1-p_e)^1-omega(e), where k(omega) is the number of connected components of the open subgraph; the case q=0 is understood as the limiting measure, equivalently Bernoulli bond percolation conditioned on being connected when such configurations exist.

Pairwise negative-correlation conjecture for the random-cluster model

For every finite graph $G=(V,E)$ , every edge-parameter vector $(p_e)_{e\in E}\in[0,1]^E$ , every $q\in[0,1]$ , and every pair of distinct edges $e,f\in E$ , the random-cluster measure $\phi_{G,\mathbf p,q}$ satisfies $\phi_{G,\mathbf p,q}(e\text{ is open},f\text{ is open})\leq \phi_{G,\mathbf p,q}(e\text{ is open})\,\phi_{G,\mathbf p,q}(f\text{ is open})$ . For $q>0$ , this measure assigns to each configuration $\omega\in\{0,1\}^E$ probability proportional to $q^{k(\omega)}\prod_{e\in E}p_e^{\omega(e)}(1-p_e)^{1-\omega(e)}$ , where $k(\omega)$ is the number of connected components of the open subgraph; the case $q=0$ is understood as the limiting measure, equivalently Bernoulli bond percolation conditioned on being connected when such configurations exist.

References

Primary source

arXiv

Additional references

Negative correlation for the random-cluster model below one on high-degree regular graphs — arXiv — Pengfei Tang, Zibo Zhang

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

The conjecture has been proved for important high-degree graph families, but it remains unsettled for arbitrary finite graphs.

The Kahn–Grimmett–Winkler conjecture asserts pairwise negative correlation in the random-cluster model for $0\leq q\leq 1$ . It remains a general finite-graph problem.

Known results

At $q=1$ , an explicit correlation formula holds for arbitrary graphs and edge weights (2025).
Conditional negative association was proved for branch-connectivity indicators on finite wired trees (2026).
Related covariance inequalities were proved under the unique infinite-volume wired measure on infinite regular trees (2026).

September 2026 high-degree regular graphs

Pengfei Tang and Zibo Zhang claim the conjecture for expander-like, high-degree regular graph sequences and establish an asymptotic reduction equating this regime with the unrestricted finite-graph problem. Their result is substantial progress but does not prove pairwise negative correlation on every finite graph.

Current status (as of September 2026): The conjecture is established at $q=1$ , in several tree settings, and for the reported high-degree regular graph families; the unrestricted finite-graph case remains open.

Sources

No solutions have been posted yet.