Dynamical random field Ising model at zero temperature
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912741558583296 |
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| author | Ding, Jian Yang, Peng Zhuang, Zijie |
| author_facet | Ding, Jian Yang, Peng Zhuang, Zijie |
| contents | In this paper, we study the evolution of the zero-temperature random field Ising model as the mean of the external field $M$ increases from $-\infty$ to $\infty$. We focus on two types of evolutions: the ground state evolution and the Glauber evolution. For the ground state evolution, we investigate the occurrence of global avalanche, a moment where a large fraction of spins flip simultaneously from minus to plus. In two dimensions, no global avalanche occurs, while in three or higher dimensions, there is a phase transition: a global avalanche happens when the noise intensity is small, but not when it is large. Additionally, we study the zero-temperature Glauber evolution, where spins are updated locally to minimize the Hamiltonian. Our results show that for small noise intensity, in dimensions $d =2$ or $3$, most spins flip around a critical time $c_d = \frac{2 \sqrt{d}}{1 + \sqrt{d}}$ (but we cannot decide whether such flipping occurs simultaneously or not). We also connect this process to polluted bootstrap percolation and solve an open problem on it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamical random field Ising model at zero temperature Ding, Jian Yang, Peng Zhuang, Zijie Probability Mathematical Physics In this paper, we study the evolution of the zero-temperature random field Ising model as the mean of the external field $M$ increases from $-\infty$ to $\infty$. We focus on two types of evolutions: the ground state evolution and the Glauber evolution. For the ground state evolution, we investigate the occurrence of global avalanche, a moment where a large fraction of spins flip simultaneously from minus to plus. In two dimensions, no global avalanche occurs, while in three or higher dimensions, there is a phase transition: a global avalanche happens when the noise intensity is small, but not when it is large. Additionally, we study the zero-temperature Glauber evolution, where spins are updated locally to minimize the Hamiltonian. Our results show that for small noise intensity, in dimensions $d =2$ or $3$, most spins flip around a critical time $c_d = \frac{2 \sqrt{d}}{1 + \sqrt{d}}$ (but we cannot decide whether such flipping occurs simultaneously or not). We also connect this process to polluted bootstrap percolation and solve an open problem on it. |
| title | Dynamical random field Ising model at zero temperature |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2410.20457 |