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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.16486 |
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| _version_ | 1866912773459410944 |
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| author | Beffara, Vincent Faipeur, Corentin Oke, Tejas |
| author_facet | Beffara, Vincent Faipeur, Corentin Oke, Tejas |
| contents | We consider the random cluster model with parameter $q<1$, for which the FKG inequalities are not valid. On the square lattice, stochastic comparison with Bernoulli percolation implies that the model is subcritical (respectively supercritical) when $p \leq q/(1+q)$ (resp. $p \geq 1/2$); in this paper, we extend these two regions, by improving the classical stochastic comparisons. Assuming the existence of the critical point, this reduces its possible range.
The proof relies on a modification of the usual Glauber dynamics of the model, which enables stochastic bounds of FK measures between two inhomegenous percolations. We also prove uniqueness of the infinite-volume measure in our extended ranges. Most of our results are valid in any dimension $d \geq 2$ and beyond hypercubic lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new bound for the critical point of the FK model for $q<1$ Beffara, Vincent Faipeur, Corentin Oke, Tejas Probability Mathematical Physics We consider the random cluster model with parameter $q<1$, for which the FKG inequalities are not valid. On the square lattice, stochastic comparison with Bernoulli percolation implies that the model is subcritical (respectively supercritical) when $p \leq q/(1+q)$ (resp. $p \geq 1/2$); in this paper, we extend these two regions, by improving the classical stochastic comparisons. Assuming the existence of the critical point, this reduces its possible range. The proof relies on a modification of the usual Glauber dynamics of the model, which enables stochastic bounds of FK measures between two inhomegenous percolations. We also prove uniqueness of the infinite-volume measure in our extended ranges. Most of our results are valid in any dimension $d \geq 2$ and beyond hypercubic lattices. |
| title | A new bound for the critical point of the FK model for $q<1$ |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2512.16486 |