Near-critical and finite-size scaling for high-dimensional lattice trees and animals
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913799008681984 |
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| author | Liu, Yucheng Slade, Gordon |
| author_facet | Liu, Yucheng Slade, Gordon |
| contents | We consider spread-out models of lattice trees and lattice animals on $\mathbb Z^d$, for $d$ above the upper critical dimension $d_{\mathrm c}=8$. We define a correlation length and prove that it diverges as $(p_c-p)^{-1/4}$ at the critical point $p_c$. Using this, we prove that the near-critical two-point function is bounded above by $C|x|^{-(d-2)}\exp[-c(p_c-p)^{1/4}|x|]$. We apply the near-critical bound to study lattice trees and lattice animals on a discrete $d$-dimensional torus (with $d > d_{\mathrm c}$) of volume $V$. For $p_c-p$ of order $V^{-1/2}$, we prove that the torus susceptibility is of order $V^{1/4}$, and that the torus two-point function behaves as $|x|^{-(d-2)} + V^{-3/4}$ and thus has a plateau of size $V^{-3/4}$. The proofs require significant extensions of previous results obtained using the lace expansion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05491 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Near-critical and finite-size scaling for high-dimensional lattice trees and animals Liu, Yucheng Slade, Gordon Probability Mathematical Physics 60K35, 82B27, 82B41, 82B43 We consider spread-out models of lattice trees and lattice animals on $\mathbb Z^d$, for $d$ above the upper critical dimension $d_{\mathrm c}=8$. We define a correlation length and prove that it diverges as $(p_c-p)^{-1/4}$ at the critical point $p_c$. Using this, we prove that the near-critical two-point function is bounded above by $C|x|^{-(d-2)}\exp[-c(p_c-p)^{1/4}|x|]$. We apply the near-critical bound to study lattice trees and lattice animals on a discrete $d$-dimensional torus (with $d > d_{\mathrm c}$) of volume $V$. For $p_c-p$ of order $V^{-1/2}$, we prove that the torus susceptibility is of order $V^{1/4}$, and that the torus two-point function behaves as $|x|^{-(d-2)} + V^{-3/4}$ and thus has a plateau of size $V^{-3/4}$. The proofs require significant extensions of previous results obtained using the lace expansion. |
| title | Near-critical and finite-size scaling for high-dimensional lattice trees and animals |
| topic | Probability Mathematical Physics 60K35, 82B27, 82B41, 82B43 |
| url | https://arxiv.org/abs/2412.05491 |