Uniform convergence of Dyson Ferrari-Spohn diffusions to the Airy line ensemble
| Autor: | Dimitrov, Evgeni, Serio, Christian |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | We consider the Dyson Ferrari-Spohn diffusion $\mathcal{X}^N = (\mathcal{X}^N_1,\dots,\mathcal{X}^N_N)$, consisting of $N$ non-intersecting Ferrari-Spohn diffusions $\mathcal{X}^N_1 > \cdots > \mathcal{X}^N_N > 0$ on $\mathbb{R}$. This object was introduced by Ioffe, Velenik, and Wachtel (2018) as a scaling limit for line ensembles of $N$ non-intersecting random walks above a hard wall with area tilts, which model certain three-dimensional interfaces in statistical physics. It was shown by Ferrari and Shlosman (2023) that as $N\to\infty$, after a spatial shift of order $N^{2/3}$ and constant rescaling in time, the top curve $\mathcal{X}^N_1$ converges to the $\mathrm{Airy}_2$ process in the sense of finite-dimensional distributions. We extend this result by showing that the full ensemble $\mathcal{X}^N$ converges with the same shift and time scaling to the Airy line ensemble in the topology of uniform convergence on compact sets. In our argument we formulate a Brownian Gibbs property with area tilts for $\mathcal{X}^N$, which we show is equivalent after a global parabolic shift to the usual Brownian Gibbs property introduced by Corwin and Hammond (2014). Comment: 20 pages |
| Databáze: | arXiv |
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