The Critical 2d Stochastic Heat Flow and Related Models
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| Format: | Preprint |
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2024
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| author | Caravenna, Francesco Sun, Rongfeng Zygouras, Nikos |
| author_facet | Caravenna, Francesco Sun, Rongfeng Zygouras, Nikos |
| contents | In these lecture notes, we review recent progress in the study of the stochastic heat equation and its discrete analogue, the directed polymer model, in spatial dimension 2. It was discovered that a phase transition emerges on an intermediate disorder scale, with Edwards-Wilkinson (Gaussian) fluctuations in the sub-critical regime. In the critical window, a unique scaling limit has been identified and named the critical 2d stochastic heat flow. This gives a meaning to the solution of the stochastic heat equation in the critical dimension 2, which lies beyond existing solution theories for singular SPDEs. We outline the proof ideas, introduce the key ingredients, and discuss related literature on disordered systems and singular SPDEs. A list of open questions is also provided. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Critical 2d Stochastic Heat Flow and Related Models Caravenna, Francesco Sun, Rongfeng Zygouras, Nikos Probability Mathematical Physics Primary: 82B44, Secondary: 35R60, 60H15, 82D60 In these lecture notes, we review recent progress in the study of the stochastic heat equation and its discrete analogue, the directed polymer model, in spatial dimension 2. It was discovered that a phase transition emerges on an intermediate disorder scale, with Edwards-Wilkinson (Gaussian) fluctuations in the sub-critical regime. In the critical window, a unique scaling limit has been identified and named the critical 2d stochastic heat flow. This gives a meaning to the solution of the stochastic heat equation in the critical dimension 2, which lies beyond existing solution theories for singular SPDEs. We outline the proof ideas, introduce the key ingredients, and discuss related literature on disordered systems and singular SPDEs. A list of open questions is also provided. |
| title | The Critical 2d Stochastic Heat Flow and Related Models |
| topic | Probability Mathematical Physics Primary: 82B44, Secondary: 35R60, 60H15, 82D60 |
| url | https://arxiv.org/abs/2412.10311 |