Existence of gradient Gibbs measures on regular trees which are not translation invariant

Florian Henning; Christof Külske

doi:10.1214/22-AAP1883

Existence of gradient Gibbs measures on regular trees which are not translation invariant

Florian Henning (Corresponding author)
, Christof Külske (Corresponding author)

Research output: Contribution to journal › Article › Academic › peer-review

5 Citations (Scopus)

Abstract

We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

Original language	English
Pages (from-to)	3010-3038
Number of pages	29
Journal	Annals of Applied Probability
Volume	33
Issue number	4
DOIs	https://doi.org/10.1214/22-AAP1883
Publication status	Published - Aug 2023
Externally published	Yes

Funding

We thank Alberto Abbondandolo for pointing out the very useful reference [7]. Moreover, we thank an anonymous referee for the insightful comments, questions and suggestions. Florian Henning was partially supported by the Research Training Group 2131 High-dimensional phenomena in probability—Fluctuations and discontinuity of German Research Council (DFG).

Funders
Deutsche Forschungsgemeinschaft

Keywords

boundary law
Gibbs measures
gradient Gibbs measures
heavy tails
regular tree
stable manifold theorem

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10.1214/22-AAP1883

Cite this

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abstract = "We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.",

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AU - Henning, Florian

AU - Külske, Christof

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N2 - We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

AB - We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

KW - boundary law

KW - Gibbs measures

KW - gradient Gibbs measures

KW - heavy tails

KW - regular tree

KW - stable manifold theorem

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JO - Annals of Applied Probability

JF - Annals of Applied Probability

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ER -

Existence of gradient Gibbs measures on regular trees which are not translation invariant

Abstract

Funding

Keywords

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New Applied Probability Findings from Ruhr-University Bochum Reported (Existence of Gradient Gibbs Measures On Regular Trees Which Are Not Translation Invariant)

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Existence of gradient Gibbs measures on regular trees which are not translation invariant

Abstract

Funding

Keywords

Access to Document

Other files and links

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Press/Media

New Applied Probability Findings from Ruhr-University Bochum Reported (Existence of Gradient Gibbs Measures On Regular Trees Which Are Not Translation Invariant)

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