Comparison of arm exponents in planar FK-percolation
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911237782110208 |
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| author | Gassmann, Loïc Manolescu, Ioan |
| author_facet | Gassmann, Loïc Manolescu, Ioan |
| contents | By the FKG inequality for FK-percolation, the probability of the alternating two-arm event is smaller than the product of the probabilities of having a primal arm and a dual arm, respectively. In this paper, we improve this inequality by a polynomial factor for critical planar FK-percolation in the continuous phase transition regime ($1 \leq q \leq 4$). In particular, we prove that if the alternating two-arm exponent $α_{01}$ and the one-arm exponents $α_0$ and $α_1$ exist, then they satisfy the strict inequality $α_{01} > α_0 + α_1$. The question was formulated by Garban and Steif in the context of exceptional times and was brought to our attention by Radhakrishnan and Tassion, who obtained the same result for planar Bernoulli percolation through different methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23013 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparison of arm exponents in planar FK-percolation Gassmann, Loïc Manolescu, Ioan Probability Mathematical Physics 82B27, 60K35, 82B43, 82B20 By the FKG inequality for FK-percolation, the probability of the alternating two-arm event is smaller than the product of the probabilities of having a primal arm and a dual arm, respectively. In this paper, we improve this inequality by a polynomial factor for critical planar FK-percolation in the continuous phase transition regime ($1 \leq q \leq 4$). In particular, we prove that if the alternating two-arm exponent $α_{01}$ and the one-arm exponents $α_0$ and $α_1$ exist, then they satisfy the strict inequality $α_{01} > α_0 + α_1$. The question was formulated by Garban and Steif in the context of exceptional times and was brought to our attention by Radhakrishnan and Tassion, who obtained the same result for planar Bernoulli percolation through different methods. |
| title | Comparison of arm exponents in planar FK-percolation |
| topic | Probability Mathematical Physics 82B27, 60K35, 82B43, 82B20 |
| url | https://arxiv.org/abs/2410.23013 |