Abstract
We consider line ensembles of non-intersecting random walks constrained by a hard wall, each tilted by the area underneath it with geometrically growing pre-factors where . This is a model for the level lines of the D SOS model above a hard wall, which itself mimics the low-temperature 3D Ising interface. A similar model with and a fixed number of curves was studied by Ioffe, Velenik, and Wachtel (2018), who derived a scaling limit as the time interval tends to infinity. Line ensembles of Brownian bridges with geometric area tilts () were studied by Caputo, Ioffe, and Wachtel (2019), and later by Dembo, Lubetzky, and Zeitouni (2022+). Their results show that as the time interval and the number of curves n tend to infinity, the top k paths converge to a limiting measure μ. In this paper we address the open problem of proving existence of a scaling limit for random walk ensembles with geometric area tilts. We prove that with mild assumptions on the jump distribution, under suitable scaling the top k paths converge to the same measure μ as followed by . We do so both in the case of bridges fixed at and of walks fixed only at .
Acknowledgments
The author would like to thank Amir Dembo for several helpful discussions regarding this work.
Citation
Christian Serio. "Scaling limit for line ensembles of random walks with geometric area tilts." Electron. J. Probab. 28 1 - 14, 2023. https://doi.org/10.1214/23-EJP1026
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