April 2023 Convergence to the thermodynamic limit for random-field random surfaces
Paul Dario
Author Affiliations +
Ann. Appl. Probab. 33(2): 1373-1395 (April 2023). DOI: 10.1214/22-AAP1844


We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient.

In this work, we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions d4 while the distribution of the surface itself converges in dimensions d5. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by Cotar and Külske (Ann. Appl. Probab. 22 (2012) 1650–1692).

Funding Statement

The research of the author was supported in part by Israel Science Foundation grants 861/15 and 1971/19 and by the European Research Council starting grant 678520 (LocalOrder).


The author would like to thank the anonymous referees who provided useful and detailed comments on an earlier version of the manuscript, as well as for suggesting a simplification in the proofs of the estimates (5) and (8).


Download Citation

Paul Dario. "Convergence to the thermodynamic limit for random-field random surfaces." Ann. Appl. Probab. 33 (2) 1373 - 1395, April 2023. https://doi.org/10.1214/22-AAP1844


Received: 1 May 2021; Revised: 1 February 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1517.82016
MathSciNet: MR4564429
Digital Object Identifier: 10.1214/22-AAP1844

Primary: 82B24 , 82B44 , 82C41

Keywords: Disordered systems , Random surfaces

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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