The Discrete Gaussian model, I. Renormalisation group flow at high temperature
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916317826646016 |
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| author | Bauerschmidt, Roland Park, Jiwoon Rodriguez, Pierre-François |
| author_facet | Bauerschmidt, Roland Park, Jiwoon Rodriguez, Pierre-François |
| contents | The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer-valued field becomes a smooth field which we then analyse using the renormalisation group method.
This paper also provides the foundation for the construction of the scaling limit of the infinite-volume gradient Gibbs state of the Discrete Gaussian model in the companion paper. Moreover, we develop all estimates for general finite-range interaction with sharp dependence on the range. We expect these estimates to prepare for a future analysis of the spread-out version of the Discrete Gaussian model at its critical temperature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_02286 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Discrete Gaussian model, I. Renormalisation group flow at high temperature Bauerschmidt, Roland Park, Jiwoon Rodriguez, Pierre-François Probability Mathematical Physics 82B20, 82B28, 60G15, 60K35 (primary) The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer-valued field becomes a smooth field which we then analyse using the renormalisation group method. This paper also provides the foundation for the construction of the scaling limit of the infinite-volume gradient Gibbs state of the Discrete Gaussian model in the companion paper. Moreover, we develop all estimates for general finite-range interaction with sharp dependence on the range. We expect these estimates to prepare for a future analysis of the spread-out version of the Discrete Gaussian model at its critical temperature. |
| title | The Discrete Gaussian model, I. Renormalisation group flow at high temperature |
| topic | Probability Mathematical Physics 82B20, 82B28, 60G15, 60K35 (primary) |
| url | https://arxiv.org/abs/2202.02286 |