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May 2021 Two-temperatures overlap distribution for the 2D discrete Gaussian free field
Michel Pain, Olivier Zindy
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Ann. Inst. H. Poincaré Probab. Statist. 57(2): 685-699 (May 2021). DOI: 10.1214/20-AIHP1091

Abstract

In this paper, we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained recently by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM’s one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.

Dans cet article, nous montrons l’absence de chaos en température pour le champ libre gaussien discret 2-dimensionnel (DGFF) en utilisant la convergence du processus extrémal obtenue récemment par Biskup et Louidor. Cela signifie que le chevauchement entre deux configurations choisies selon les mesures de Gibbs à différentes températures a une distribution non triviale à la limite. Alors que cette distribution est identique à celle du modèle d’énergies aléatoires (REM) dans le cas de configurations choisies à la même température, nous montrons ici que ces distributions sont distinctes dans le cas de températures différentes. Plus précisément, nous démontrons que le chevauchement moyen entre deux configurations choisies selon les mesures de Gibbs à différentes températures pour le DGFF est strictement plus petit que celui obtenu dans le cas du REM. Ainsi, bien qu’aucun de ces modèles ne présente de chaos en température, on peut dire que le DGFF est moins chaotique en température que le REM.

Acknowledgements

We thank Antonio Auffinger for suggesting the question of chaos in temperature for the two-dimensional discrete Gaussian free field and Bernard Derrida for very stimulating discussions. The authors also wish to thank the referees for useful comments that improved the presentation of the paper.

Citation

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Michel Pain. Olivier Zindy. "Two-temperatures overlap distribution for the 2D discrete Gaussian free field." Ann. Inst. H. Poincaré Probab. Statist. 57 (2) 685 - 699, May 2021. https://doi.org/10.1214/20-AIHP1091

Information

Received: 6 December 2018; Revised: 13 July 2020; Accepted: 29 July 2020; Published: May 2021
First available in Project Euclid: 13 May 2021

Digital Object Identifier: 10.1214/20-AIHP1091

Subjects:
Primary: 60G15, 60G70, 82D30

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 2 • May 2021
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