The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866917096661712896 |
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| author | Chatzigeorgiou, Georgiana Morfe, Peter Otto, Felix Wang, Lihan |
| author_facet | Chatzigeorgiou, Georgiana Morfe, Peter Otto, Felix Wang, Lihan |
| contents | We analyze the large-time asymptotics of a passive tracer with drift equal to the curl of the Gaussian free field in two dimensions with ultra-violet cut-off at scale unity. We prove that the mean-squared displacement scales like $t \sqrt{\ln t}$, as predicted in the physics literature and recently almost proved by the work of Cannizzaro, Haunschmidt-Sibitz, and Toninelli (2022), which uses mathematical-physics type analysis in Fock space. Our approach involves studying the effective diffusivity $λ_{L}$ of the process with an infra-red cut-off at scale $L$, and is based on techniques from stochastic homogenization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_14244 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off Chatzigeorgiou, Georgiana Morfe, Peter Otto, Felix Wang, Lihan Probability Analysis of PDEs 35B27, 35R60, 60J60 We analyze the large-time asymptotics of a passive tracer with drift equal to the curl of the Gaussian free field in two dimensions with ultra-violet cut-off at scale unity. We prove that the mean-squared displacement scales like $t \sqrt{\ln t}$, as predicted in the physics literature and recently almost proved by the work of Cannizzaro, Haunschmidt-Sibitz, and Toninelli (2022), which uses mathematical-physics type analysis in Fock space. Our approach involves studying the effective diffusivity $λ_{L}$ of the process with an infra-red cut-off at scale $L$, and is based on techniques from stochastic homogenization. |
| title | The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off |
| topic | Probability Analysis of PDEs 35B27, 35R60, 60J60 |
| url | https://arxiv.org/abs/2212.14244 |