Large scale regularity and correlation length for almost length-minimizing random curves in the plane
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| Format: | Preprint |
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2024
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| _version_ | 1866917877441888256 |
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| author | Ried, Tobias Wagner, Christian |
| author_facet | Ried, Tobias Wagner, Christian |
| contents | We consider a model of random curves in the plane related to the large-scale behavior of the Random Field Ising Model (RFIM) at temperature zero in two space dimensions. Our work is motivated by attempts to quantify the Imry--Ma phenomenon concerning the rounding of the phase transition by quenched disorder, and connects to recent advances regarding the decay of correlations in the RFIM.
We study a continuum model of minimal surfaces in two space dimensions subject to an external, quenched random field, and restrict ourselves to isotropic surface integrands. The random fields we consider behave like white noise on large scales with an ultra-violet regularization reminiscent of the lattice structure of the RFIM.
We give a finer description of the minimizer below the length scale starting from which the influence of boundary conditions is suppressed with a given probability, which has recently been shown to be of order $\exp(\varepsilon^{-\frac{4}{3}})$ in the amplitude $\varepsilon>0$ of the noise. More precisely, we prove flatness of the phase boundaries on scales up to order $\exp(\varepsilon^{-\frac{4}{13}})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_17625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large scale regularity and correlation length for almost length-minimizing random curves in the plane Ried, Tobias Wagner, Christian Mathematical Physics Analysis of PDEs Probability 82B44, 49Q05, 60K35 We consider a model of random curves in the plane related to the large-scale behavior of the Random Field Ising Model (RFIM) at temperature zero in two space dimensions. Our work is motivated by attempts to quantify the Imry--Ma phenomenon concerning the rounding of the phase transition by quenched disorder, and connects to recent advances regarding the decay of correlations in the RFIM. We study a continuum model of minimal surfaces in two space dimensions subject to an external, quenched random field, and restrict ourselves to isotropic surface integrands. The random fields we consider behave like white noise on large scales with an ultra-violet regularization reminiscent of the lattice structure of the RFIM. We give a finer description of the minimizer below the length scale starting from which the influence of boundary conditions is suppressed with a given probability, which has recently been shown to be of order $\exp(\varepsilon^{-\frac{4}{3}})$ in the amplitude $\varepsilon>0$ of the noise. More precisely, we prove flatness of the phase boundaries on scales up to order $\exp(\varepsilon^{-\frac{4}{13}})$. |
| title | Large scale regularity and correlation length for almost length-minimizing random curves in the plane |
| topic | Mathematical Physics Analysis of PDEs Probability 82B44, 49Q05, 60K35 |
| url | https://arxiv.org/abs/2412.17625 |