Tightness of Bernoulli Gibbsian line ensembles
| Autor: | Dimitrov, Evgeni, Fang, Xiang, Fesser, Lukas, Serio, Christian, Teitler, Carson, Wang, Angela, Zhu, Weitao |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | A Bernoulli Gibbsian line ensemble $\mathfrak{L} = (L_1, \dots, L_N)$ is the law of the trajectories of $N-1$ independent Bernoulli random walkers $L_1, \dots, L_{N-1}$ with possibly random initial and terminal locations that are conditioned to never cross each other or a given random up-right path $L_N$ (i.e. $L_1 \geq \cdots \geq L_N$). In this paper we investigate the asymptotic behavior of sequences of Bernoulli Gibbsian line ensembles $\mathfrak{L}^N = (L^N_1, \dots, L^N_N)$ when the number of walkers $N$ tends to infinity. We prove that if one has mild but uniform control of the one-point marginals of the lowest-indexed (or top) curves $L_1^N$ then the sequence $\mathfrak{L}^N$ is tight in the space of line ensembles. Furthermore, we show that if the top curves $L_1^N$ converge in the finite dimensional sense to the parabolic Airy$_2$ process then $\mathfrak{L}^N$ converge to the parabolically shifted Airy line ensemble. Comment: 85 pages, 5 figures. Fixed a few minor mistakes in the second version |
| Databáze: | arXiv |
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