Near-critical Ornstein--Zernike theory for the planar random-cluster model
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| Format: | Preprint |
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2025
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| author | D'Alimonte, Lucas Manolescu, Ioan |
| author_facet | D'Alimonte, Lucas Manolescu, Ioan |
| contents | We develop an Ornstein--Zernike theory for the two-dimensional random-cluster model with $1 \leq q <4$ that also applies in its near-critical regime. In particular, we prove an asymptotic formula for the two-point function which holds uniformly for~$p < p_c$ and blends the subcritical and near-critical behaviours of the model.
The analysis is carried out by studying the renewal properties of a subcritical percolation cluster, \emph{at the scale of the correlation length}. More precisely, we explore sequentially the cluster in a given direction, by slices of thickness comparable to the correlation length. We show that this exploration satisfies the properties of a {\em killed Markov renewal process} -- a class of processes that may be analysed independently and have Brownian behaviour. In addition to the two-point function estimate, we derive other consequences of the Ornstein--Zernike theory such as an invariance principle for the rescaled cluster and the strict convexity of the inverse correlation length -- all at the scale of the correlation length, uniformly in~$p<p_c$.
Finally, our approach differs from that of earlier papers of Campanino, Ioffe, Velenik and others, with the cluster being dynamically explored rather than constructed from its diamond decomposition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_13648 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Near-critical Ornstein--Zernike theory for the planar random-cluster model D'Alimonte, Lucas Manolescu, Ioan Probability Mathematical Physics 60K35, 82B20, 82B43 We develop an Ornstein--Zernike theory for the two-dimensional random-cluster model with $1 \leq q <4$ that also applies in its near-critical regime. In particular, we prove an asymptotic formula for the two-point function which holds uniformly for~$p < p_c$ and blends the subcritical and near-critical behaviours of the model. The analysis is carried out by studying the renewal properties of a subcritical percolation cluster, \emph{at the scale of the correlation length}. More precisely, we explore sequentially the cluster in a given direction, by slices of thickness comparable to the correlation length. We show that this exploration satisfies the properties of a {\em killed Markov renewal process} -- a class of processes that may be analysed independently and have Brownian behaviour. In addition to the two-point function estimate, we derive other consequences of the Ornstein--Zernike theory such as an invariance principle for the rescaled cluster and the strict convexity of the inverse correlation length -- all at the scale of the correlation length, uniformly in~$p<p_c$. Finally, our approach differs from that of earlier papers of Campanino, Ioffe, Velenik and others, with the cluster being dynamically explored rather than constructed from its diamond decomposition. |
| title | Near-critical Ornstein--Zernike theory for the planar random-cluster model |
| topic | Probability Mathematical Physics 60K35, 82B20, 82B43 |
| url | https://arxiv.org/abs/2510.13648 |