High-dimensional long-range statistical mechanical models have random walk correlation functions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918258182979584 |
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| author | Liu, Yucheng |
| author_facet | Liu, Yucheng |
| contents | We consider long-range percolation, Ising model, and self-avoiding walk on $\mathbb{Z}^d$, with couplings decaying like $|x|^{-(d+α)}$ where $0 < α\le 2$, above the upper critical dimensions. In the spread-out setting where the lace expansion applies, we show that the two-point function for each of these models exactly coincides with a random walk two-point function, up to a constant prefactor. Using this, for $0<α< 2$, we prove upper and lower bounds of the form $|x|^{-(d-α)} \min\{ 1, (p_c - p)^{-2} |x|^{-2α} \}$ for the two-point function near the critical point $p_c$. For $α=2$, we obtain a similar upper bound with logarithmic corrections. We also give a simple proof of the convergence of the lace expansion, assuming diagrammatic estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12104 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | High-dimensional long-range statistical mechanical models have random walk correlation functions Liu, Yucheng Probability Mathematical Physics 60K35, 82B20, 82B27, 82B41, 82B43 We consider long-range percolation, Ising model, and self-avoiding walk on $\mathbb{Z}^d$, with couplings decaying like $|x|^{-(d+α)}$ where $0 < α\le 2$, above the upper critical dimensions. In the spread-out setting where the lace expansion applies, we show that the two-point function for each of these models exactly coincides with a random walk two-point function, up to a constant prefactor. Using this, for $0<α< 2$, we prove upper and lower bounds of the form $|x|^{-(d-α)} \min\{ 1, (p_c - p)^{-2} |x|^{-2α} \}$ for the two-point function near the critical point $p_c$. For $α=2$, we obtain a similar upper bound with logarithmic corrections. We also give a simple proof of the convergence of the lace expansion, assuming diagrammatic estimates. |
| title | High-dimensional long-range statistical mechanical models have random walk correlation functions |
| topic | Probability Mathematical Physics 60K35, 82B20, 82B27, 82B41, 82B43 |
| url | https://arxiv.org/abs/2502.12104 |