Absence of Shift-Invariant Gibbs States (Delocalisation) for One-Dimensional \(\pmb {\mathbb {Z}}\)-Valued Fields With Long-Range Interactions

We show that a modification of the proof of our paper Coquille et al. (J. Stat. Phys. 172(5), 1210–1222 (2018)), in the spirit of Fröhlich and Pfister (Commun. Math. Phys. 81, 277–298 (1981)), shows delocalisation in the long-range Discrete Gaussian Chain, and generalisations thereof, for any decay power \(\alpha >2\) and at all temperatures. The argument proceeds by contradiction: any shift-invariant and localised measure (in the \(L^1\) sense), is a convex combination of ergodic localised measures. But the latter cannot exist: on one hand, by the ergodic theorem, the average of the field over growing boxes would be almost surely bounded ; on the other hand the measure would be absolutely continuous with respect to its height-shifted translates, as a simple relative entropy computation shows. This leads to a contradiction and answers, in a non-quantitative way, an open question stated in a recent paper of C. Garban (Invisibility of the integers for the discrete Gaussian Chain via a caffarelli-silvestre extension of the discrete fractional laplacian. Preprint arXiv:2312.04536v2, (2023)).

The manuscript has no associated data.

The manuscript has no associated code.

1. The constant \(C_3\) in Equation (13) of the proof below would be replaced by \(C_3n\).

2. this follows from the fact that \(\nu \) is a Gibbs measure.

3. using \(|a+b|^p\le 2^{p-1}(|a|^p+|b|^p)\) for any \(p\ge 1\).

4. \(RE(\mu |\nu )\le \lim _{n\rightarrow \infty }RE(\mu _n|\nu _n)\) whenever \(\mu _n\rightarrow \mu \) and \(\nu _n\rightarrow \nu \) weakly.

We thank Christophe Garban for a helpful feedback.

Research of A.L.N. and L.C. have been supported by the CNRS IRP (International Research Project) EURANDOM “Random Graph, Statistical Mechanics and Networks” and by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d’Avenir. W.M.R. is funded by Vidi grant VI.Vidi.213.112 from the Dutch Research Council.

Authors and Affiliations

1. Univ. Grenoble Alpes, CNRS, Institut Fourier, Grenoble, F-38000, France

   Loren Coquille

2. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Nijenborgh 9, Groningen, 9747AG, The Netherlands

   Aernout van Enter

3. LAMA UMR CNRS 8050, UPEC, Université Paris-Est, 61 Avenue du Général de Gaulle, Créteil cedex, 94010, France

   Arnaud Le Ny

4. Utrecht University, Hans Freudenthalgebouw Budapestlaan 6, Utrecht, 3584 CD, The Netherlands

   Wioletta M. Ruszel

Corresponding author

Correspondence to Loren Coquille.

Conflict of interest

Aernout van Enter is an Associate Editor of J. Stat. Phys.. L.C., A.L.N. and W.M.R. have no Conflict of interest to declare that are relevant to the content of this article.

Consent for publication

Not applicable.

Ethics approval and consent to participate

Not applicable.

Materials availability

The manuscript has no associated materials.

Financial interests

The authors have no relevant financial or non-financial interests to disclose.

Communicated by Yvan Velenik.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions