October 2021 Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees
Florian Henning, Christof Külske
Author Affiliations +
Ann. Appl. Probab. 31(5): 2284-2310 (October 2021). DOI: 10.1214/20-AAP1647


We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with d+1 neighbors. We first provide general conditions in terms of the relevant p-norms of the associated transfer operator Q which ensure the existence of a countable family of proper Gibbs measures, describing localization at different heights. Next we prove existence of delocalized gradient Gibbs measures, under natural conditions on Q. We show that the two conditions can be fulfilled at the same time, which then implies coexistence of both types of measures for large classes of models including the SOS-model, and heavy-tailed models arising for instance for potentials of logarithmic growth.

Funding Statement

Florian Henning is partially supported by the Research Training Group 2131 High-dimensional phenomena in probability-Fluctuations and discontinuity of German Research Council (DFG).


The authors thank an anonymous referee for clarifying remarks and suggestions.


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Florian Henning. Christof Külske. "Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees." Ann. Appl. Probab. 31 (5) 2284 - 2310, October 2021. https://doi.org/10.1214/20-AAP1647


Received: 1 February 2020; Revised: 1 August 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

MathSciNet: MR4332697
zbMATH: 1480.82008
Digital Object Identifier: 10.1214/20-AAP1647

Primary: 82B26
Secondary: 60K35

Keywords: boundary law , delocalization , Gibbs measures , gradient Gibbs measures , heavy tails , Localization , regular tree

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.31 • No. 5 • October 2021
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