Negative correlation for the random-cluster model below one on high-degree regular graphs
Abstract
The random-cluster model with cluster parameter 0<q<1 is conjectured to exhibit negative dependence, but even pairwise negative correlation between distinct edges remains open on general finite graphs. We first establish a reduction showing that the pairwise conjecture is equivalent to its restriction to regular graph sequences of diverging degrees (dn), even for edge pairs at distance o(dn/log dn). We then prove strict pairwise negative correlation for a broad class of high-degree regular graph sequences satisfying a two-scale edge-isoperimetric condition. More precisely, for every fixed p∈(0,1), all sufficiently large graphs in the sequence have strictly negative covariance between any two distinct edge indicators at distance o(dn/log dn), uniformly in q∈[0,1) and in the choice of the two edges. In particular, the result includes the q=0 endpoint, corresponding to Bernoulli bond percolation conditioned to be connected. The isoperimetric hypothesis is satisfied by high-degree expander families and, with probability tending to one, by uniformly random regular graphs of diverging degree; its two-scale form also allows product geometries such as hypercubes and fixed-side high-dimensional discrete tori. The proof is based on a polymer representation and cluster expansion, together with a geometric identification of the leading contribution to the two-edge correlation.
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BibTeX
@misc{tang2026negative,
title = {{Negative correlation for the random-cluster model below one on high-degree regular graphs}},
author = {Pengfei Tang and Zibo Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.11578},
eprint = {2609.11578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11578},
}
RIS
TY - UNPB
TI - Negative correlation for the random-cluster model below one on high-degree regular graphs
AU - Pengfei Tang
AU - Zibo Zhang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.11578
UR - https://arxiv.org/abs/2609.11578
AB - The random-cluster model with cluster parameter 0<q<1 is conjectured to exhibit negative dependence, but even pairwise negative correlation between distinct edges remains open on general finite graphs. We first establish a reduction showing that the pairwise conjecture is equivalent to its restriction to regular graph sequences of diverging degrees (d_n), even for edge pairs at distance o(d_n/log d_n). We then prove strict pairwise negative correlation for a broad class of high-degree regular graph sequences satisfying a two-scale edge-isoperimetric condition. More precisely, for every fixed p∈(0,1), all sufficiently large graphs in the sequence have strictly negative covariance between any two distinct edge indicators at distance o(d_n/log d_n), uniformly in q∈[0,1) and in the choice of the two edges. In particular, the result includes the q=0 endpoint, corresponding to Bernoulli bond percolation conditioned to be connected. The isoperimetric hypothesis is satisfied by high-degree expander families and, with probability tending to one, by uniformly random regular graphs of diverging degree; its two-scale form also allows product geometries such as hypercubes and fixed-side high-dimensional discrete tori. The proof is based on a polymer representation and cluster expansion, together with a geometric identification of the leading contribution to the two-edge correlation.
ER -