Open Access
2021 Tightness of Bernoulli Gibbsian line ensembles
Evgeni Dimitrov, Xiang Fang, Lukas Fesser, Christian Serio, Carson Teitler, Angela Wang, Weitao Zhu
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Electron. J. Probab. 26: 1-93 (2021). DOI: 10.1214/21-EJP698


A Bernoulli Gibbsian line ensemble L=(L1,,LN) is the law of the trajectories of N1 independent Bernoulli random walkers L1,,LN1 with possibly random initial and terminal locations that are conditioned to never cross each other or a given random up-right path LN (i.e. L1LN). In this paper we investigate the asymptotic behavior of sequences of Bernoulli Gibbsian line ensembles LN=(L1N,,LNN) 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 L1N then the sequence LN is tight in the space of line ensembles. Furthermore, we show that if the top curves L1N converge in the finite dimensional sense to the parabolic Airy2 process then LN converge to the parabolic Airy line ensemble.


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.


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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.


Received: 28 February 2021; Accepted: 1 September 2021; Published: 2021
First available in Project Euclid: 25 November 2021

arXiv: 2011.04478
Digital Object Identifier: 10.1214/21-EJP698

Primary: 60J65 , 82B41

Keywords: avoiding random walks , Brownian motion , Gibbsian line ensembles

Vol.26 • 2021
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