New lower bounds for the (near) critical Ising and $φ^4$ models' two-point functions
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| Format: | Preprint |
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2024
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| _version_ | 1866909453334347776 |
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| author | Duminil-Copin, Hugo Panis, Romain |
| author_facet | Duminil-Copin, Hugo Panis, Romain |
| contents | We study the nearest-neighbour Ising and $φ^4$ models on $\mathbb Z^d$ with $d\geq 3$ and obtain new lower bounds on their two-point functions at (and near) criticality. Together with the classical infrared bound, these bounds turn into up-to constant estimates when $d\geq 5$. When $d=4$, we obtain an ''almost'' sharp lower bound corrected by a logarithmic factor. As a consequence of these results, we show that $η=0$ and $ν=1/2$ when $d\geq 4$, where $η$ is the critical exponent associated with the decay of the model's two-point function at criticality and $ν$ is the critical exponent of the correlation length $ξ(β)$. When $d=3$, we improve previous results and obtain that $η\leq 1/2$. As a byproduct of our proofs, we also derive the blow-up at criticality of the so-called bubble diagram when $d=3,4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_05700 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New lower bounds for the (near) critical Ising and $φ^4$ models' two-point functions Duminil-Copin, Hugo Panis, Romain Probability Mathematical Physics 60K35, 82B20, 82B27 We study the nearest-neighbour Ising and $φ^4$ models on $\mathbb Z^d$ with $d\geq 3$ and obtain new lower bounds on their two-point functions at (and near) criticality. Together with the classical infrared bound, these bounds turn into up-to constant estimates when $d\geq 5$. When $d=4$, we obtain an ''almost'' sharp lower bound corrected by a logarithmic factor. As a consequence of these results, we show that $η=0$ and $ν=1/2$ when $d\geq 4$, where $η$ is the critical exponent associated with the decay of the model's two-point function at criticality and $ν$ is the critical exponent of the correlation length $ξ(β)$. When $d=3$, we improve previous results and obtain that $η\leq 1/2$. As a byproduct of our proofs, we also derive the blow-up at criticality of the so-called bubble diagram when $d=3,4$. |
| title | New lower bounds for the (near) critical Ising and $φ^4$ models' two-point functions |
| topic | Probability Mathematical Physics 60K35, 82B20, 82B27 |
| url | https://arxiv.org/abs/2404.05700 |