Abstract
A Bernoulli Gibbsian line ensemble is the law of the trajectories of independent Bernoulli random walkers with possibly random initial and terminal locations that are conditioned to never cross each other or a given random up-right path (i.e. ). In this paper we investigate the asymptotic behavior of sequences of Bernoulli Gibbsian line ensembles when the number of walkers 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 then the sequence is tight in the space of line ensembles. Furthermore, we show that if the top curves converge in the finite dimensional sense to the parabolic Airy process then converge to the parabolic Airy line ensemble.
Acknowledgments
This project was initiated during the summer REU program at Columbia University in 2020 and we thank the organizer from the Mathematics department, Michael Woodbury, for this wonderful opportunity. E.D. is partially supported by the Minerva Foundation Fellowship.
Citation
Evgeni Dimitrov. Xiang Fang. Lukas Fesser. Christian Serio. Carson Teitler. Angela Wang. Weitao Zhu. "Tightness of Bernoulli Gibbsian line ensembles." Electron. J. Probab. 26 1 - 93, 2021. https://doi.org/10.1214/21-EJP698
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