Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+)
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| Format: | Preprint |
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2024
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| _version_ | 1866911125516320768 |
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| author | Abuzaid, Osama Healey, Vivian Olsiewski Peltola, Eveliina |
| author_facet | Abuzaid, Osama Healey, Vivian Olsiewski Peltola, Eveliina |
| contents | We show a finite-time large deviation principle (LDP) for "Dyson type" diffusion processes, including Dyson Brownian motion on the circle, for a fixed number of particles as the coupling parameter $β=8/κ$ tends to $\infty$. We also characterize the large-time behavior of finite-energy and zero-energy systems. Interestingly, the latter correspond to the Calogero-Moser-Sutherland integrable system.
We use these results to derive an LDP in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$_κ$, as $κ\to 0$, with good rate function being the multiradial Loewner energy. Here, the main difficulty is that the curves have a common target point, preventing the configurational (global) approach. Our proof thus requires topological results in Loewner theory: using a derivative estimate for the radial Loewner map in terms of the energy of its driving function, we show that finite-energy multiradial Loewner hulls are disjoint unions of simple curves, except possibly at their common endpoint. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_13762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+) Abuzaid, Osama Healey, Vivian Olsiewski Peltola, Eveliina Probability Mathematical Physics We show a finite-time large deviation principle (LDP) for "Dyson type" diffusion processes, including Dyson Brownian motion on the circle, for a fixed number of particles as the coupling parameter $β=8/κ$ tends to $\infty$. We also characterize the large-time behavior of finite-energy and zero-energy systems. Interestingly, the latter correspond to the Calogero-Moser-Sutherland integrable system. We use these results to derive an LDP in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$_κ$, as $κ\to 0$, with good rate function being the multiradial Loewner energy. Here, the main difficulty is that the curves have a common target point, preventing the configurational (global) approach. Our proof thus requires topological results in Loewner theory: using a derivative estimate for the radial Loewner map in terms of the energy of its driving function, we show that finite-energy multiradial Loewner hulls are disjoint unions of simple curves, except possibly at their common endpoint. |
| title | Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+) |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2407.13762 |