Hierarchy of KPZ limits arising from directed random walk models in random media
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908682451681280 |
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| author | Parekh, Shalin |
| author_facet | Parekh, Shalin |
| contents | We consider a generalized model of random walk in dynamical random environment, and we show that the multiplicative-noise stochastic heat equation (SHE) describes the fluctuations of the quenched density at a certain precise location in the tail. The distribution of transition kernels is fixed rather than changing under the diffusive rescaling of space-time, i.e., there is no critical tuning of the model parameters when scaling to the stochastic PDE limit. The proof is done by pushing the methods developed in [arxiv 2304.14279, arXiv 2311.09151] to their maximum, substantially weakening the assumptions and obtaining fairly sharp conditions under which one expects to see the SHE arise in a wide variety of random walk models in random media. In particular we are able to get rid of conditions such as nearest-neighbor interaction as well as spatial independence of quenched transition kernels. Moreover, we observe an entire hierarchy of moderate deviation exponents at which the SHE can be found, confirming a physics prediction of J. Hass. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06073 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hierarchy of KPZ limits arising from directed random walk models in random media Parekh, Shalin Probability Mathematical Physics We consider a generalized model of random walk in dynamical random environment, and we show that the multiplicative-noise stochastic heat equation (SHE) describes the fluctuations of the quenched density at a certain precise location in the tail. The distribution of transition kernels is fixed rather than changing under the diffusive rescaling of space-time, i.e., there is no critical tuning of the model parameters when scaling to the stochastic PDE limit. The proof is done by pushing the methods developed in [arxiv 2304.14279, arXiv 2311.09151] to their maximum, substantially weakening the assumptions and obtaining fairly sharp conditions under which one expects to see the SHE arise in a wide variety of random walk models in random media. In particular we are able to get rid of conditions such as nearest-neighbor interaction as well as spatial independence of quenched transition kernels. Moreover, we observe an entire hierarchy of moderate deviation exponents at which the SHE can be found, confirming a physics prediction of J. Hass. |
| title | Hierarchy of KPZ limits arising from directed random walk models in random media |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2401.06073 |