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February 2024 On the limiting law of line ensembles of Brownian polymers with geometric area tilts
Amir Dembo, Eyal Lubetzky, Ofer Zeitouni
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 113-125 (February 2024). DOI: 10.1214/22-AIHP1284


We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the level lines of the (2+1)d sos model above a hard wall, was studied in two works from 2019 by Caputo, Ioffe and Wachtel. In those works, the tightness of the law of the top k paths, for any fixed k, was established under either zero or free boundary conditions, which in the former setting implied the existence of a limit via a monotonicity argument. Here we address the open problem of existence of a limit under free boundary conditions: we prove that as the interval length, followed by the number of paths, go to , the top k paths converge to the same limit as in the zero boundary case, as conjectured by Caputo, Ioffe and Wachtel.

Nous étudions l’ensemble de lignes déterminé par des mouvements Browniens non-intersectant au-dessus d’un mur solide. Ce modèle, qui imite les lignes de niveaux du modèle (2+1)d sos au-dessus d’un mur, a été étudié en 2019 par Caputo, Ioffe et Wachtel. Dans ces travaux, la tension de la loi des k lignes hautes, pour chaque k fixe, a été obtenue sous des conditions nulles au bord ou des conditions libres au bord. Dans le premier cas, ca implique l’existence d’une limite par un argument de monotonicité. Nous abordons ici le problème ouvert d’existence d’une limite sous des conditions libres au bord : nous démontrons que quand la longueur de l’intervalle, suivi par le nombre de lignes, tend vers l’infinie, les k lignes hautes convergent vers la même limite que dans le cas de conditions nulles au bord, comme conjecturé par Caputo, Ioffe et Wachtel.

Funding Statement

A.D. was supported in part by NSF grant DMS-1954337.
E.L. was supported in part by NSF grants DMS-1812095 and DMS-2054833.
O.Z. was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 692452).
This research was further supported in part by BSF grant 2018088.


Dedicated to the memory of Dima Ioffe


We thank Ivan Corwin for bringing to our attention the paper [10]. We thank the referees for a careful reading of the manuscript and their comments.


Download Citation

Amir Dembo. Eyal Lubetzky. Ofer Zeitouni. "On the limiting law of line ensembles of Brownian polymers with geometric area tilts." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 113 - 125, February 2024.


Received: 5 January 2022; Revised: 7 May 2022; Accepted: 10 May 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718376
Digital Object Identifier: 10.1214/22-AIHP1284

Primary: 60J60
Secondary: 82B24

Keywords: Brownian polymers , line ensembles , SOS model

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré

Vol.60 • No. 1 • February 2024
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