February 2024 Short- and long-time path tightness of the continuum directed random polymer
Sayan Das, Weitao Zhu
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 343-372 (February 2024). DOI: 10.1214/22-AIHP1334


We consider the point-to-point continuum directed random polymer (CDRP) model that arises as a scaling limit from (1+1)-dimensional directed polymers in the intermediate disorder regime. We show that the annealed law of a point-to-point CDRP of length t converges to the Brownian bridge under diffusive scaling when t0. In case that t is large, we show that the transversal fluctuations of point-to-point CDRP are governed by the 2/3 exponent. More precisely, as t tends to infinity, we prove tightness of the annealed path measures of point-to-point CDRP of length t upon scaling the length by t and fluctuations of paths by t2/3. The 2/3 exponent is tight such that the one-point distribution of the rescaled paths converges to the geodesic of the directed landscape. This point-wise convergence can be enhanced to process-level modulo a conjecture. Our short- and long-time tightness results also extend to point-to-line CDRP. In the course of proving our main results, we establish quantitative versions of quenched modulus of continuity estimates for long-time CDRP which are of independent interest.

Nous considérons le modèle de polymère dirigé continu point à point (CDRP) qui se présente comme une limite d’échelle des polymères dirigés de dimension (1+1) dans le régime du désordre intermédiaire. Nous montrons que la loi recuite d’un CDRP point à point de longueur t sous une renormalisation diffusive converge vers le pont brownien lorsque t0. Dans le cas où t est grand, nous montrons que les fluctuations transversales d’un CDRP point à point sont régies par l’exposant 2/3. Plus précisément, lorsque t tend vers l’infini, nous prouvons la tension de la mesure recuite des trajectoires de CDRP point-à-point de longueur t en renormalisant la longueur par t et les fluctuations des trajectoires par t2/3. L’exposant 2/3 est tendu de telle sorte que la loi en un point des trajectoires renormalisées converge vers la géodésique du paysage dirigé. Cette convergence ponctuelle peut être améliorée au niveau du processus modulo une conjecture. Nos résultats de tension à temps court et à temps long s’étendent également au CDRP point-à-ligne. Au cours de la démonstration de nos principaux résultats, nous établissons des versions quantitatives des estimations trempées du module de continuité pour le CDRP à temps long, qui présentent un intérêt indépendant.

Funding Statement

The project was initiated during the authors’ participation in the ‘Universality and Integrability in Random Matrix Theory and Interacting Particle Systems’ research program hosted by the Mathematical Sciences Research Institute (MSRI) in Berkeley, California in fall 2021. The authors thank the program organizers for their hospitality and acknowledge the support from NSF DMS-1928930.


The authors thank Ivan Corwin, Shirshendu Ganguly and Promit Ghosal for suggesting the problem and useful discussions. The authors are grateful to the anonymous referee for their careful reading and many valuable comments on improving our manuscript.


Download Citation

Sayan Das. Weitao Zhu. "Short- and long-time path tightness of the continuum directed random polymer." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 343 - 372, February 2024. https://doi.org/10.1214/22-AIHP1334


Received: 19 May 2022; Revised: 31 August 2022; Accepted: 3 October 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718384
Digital Object Identifier: 10.1214/22-AIHP1334

Primary: 60K37 , 82B21
Secondary: 82D60

Keywords: Brownian bridge , Directed polymer , Kardar–Parisi–Zhang equation , Stochastic heat equation

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


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Vol.60 • No. 1 • February 2024
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