A phase transition for the two-dimensional random field Ising/FK-Ising model
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908426238427136 |
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| author | Hao, Chenxu Huang, Fenglin Xia, Aoteng |
| author_facet | Hao, Chenxu Huang, Fenglin Xia, Aoteng |
| contents | We study the total variation (TV) distance between the laws of the 2D Ising/FK-Ising model in a box of side-length $N$ with and without an i.i.d.\ Gaussian external field with variance $ε^2$. Letting the external field strength $ε= ε(N)$ depend on the size of the box, we derive a phase transition for each model depending on the order of $ε(N)$. For the random field Ising model, the critical order for $ε$ is $N^{-1}$. For the random field FK-Ising model, the critical order depends on the temperature regime: for $T>T_c$, $T=T_c$ and $T\in (0, T_c)$ the critical order for $ε$ is, respectively, $N^{-\frac{1}{2}}$, $N^{-\frac{15}{16}}$ and $N^{-1}$. In each case, as $N \to \infty$ the TV distance under consideration converges to $1$ when $ε$ is above the respective critical order and converges to $0$ when below. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16268 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A phase transition for the two-dimensional random field Ising/FK-Ising model Hao, Chenxu Huang, Fenglin Xia, Aoteng Probability Mathematical Physics 60K35, 82B44 We study the total variation (TV) distance between the laws of the 2D Ising/FK-Ising model in a box of side-length $N$ with and without an i.i.d.\ Gaussian external field with variance $ε^2$. Letting the external field strength $ε= ε(N)$ depend on the size of the box, we derive a phase transition for each model depending on the order of $ε(N)$. For the random field Ising model, the critical order for $ε$ is $N^{-1}$. For the random field FK-Ising model, the critical order depends on the temperature regime: for $T>T_c$, $T=T_c$ and $T\in (0, T_c)$ the critical order for $ε$ is, respectively, $N^{-\frac{1}{2}}$, $N^{-\frac{15}{16}}$ and $N^{-1}$. In each case, as $N \to \infty$ the TV distance under consideration converges to $1$ when $ε$ is above the respective critical order and converges to $0$ when below. |
| title | A phase transition for the two-dimensional random field Ising/FK-Ising model |
| topic | Probability Mathematical Physics 60K35, 82B44 |
| url | https://arxiv.org/abs/2503.16268 |