Strong Spatial Mixing in Homomorphism Spaces

Briceño, Raimundo; Pavlov, Ronnie

doi:10.1137/16m1066178

Cited by 13 publications

(18 citation statements)

References 25 publications

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“…Let us briefly introduce graph homomorphisms and hom‐shifts. For background and more, we refer, for instance, to [6, 7, 11]. In the category of graphs, a homomorphism is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices.…”

Section: Universality For Graph Homomorphismsmentioning

confidence: 99%

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Chandgotia

Meyerovitch²

2021

Proc. London Math. Soc.

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A Borel Zd dynamical system (X,S) is ‘almost Borel universal’ if any free Borel Zd dynamical system (Y,T) of strictly lower entropy is isomorphic to a Borel subsystem of (X,S), after removing a null set. We obtain and exploit a new sufficient condition for a topological Zd dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a ‘generic’ homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non‐uniform specification implies almost Borel universality, and that 3‐colorings in Zd and dimers in Z2 are almost Borel universal.

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Section: Universality For Graph Homomorphismsmentioning

confidence: 99%

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Chandgotia

Meyerovitch²

2021

Proc. London Math. Soc.

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“…Mixing properties have been intensively studied in statistical physics and related areas (see [2,6,8,10,17,42]), and are usually applied when the set of particles in G is very large or infinite. In this case, it can be very useful to be able to "glue" together partial homomorphisms, provided their domains are far from each other.…”

Section: Mixingmentioning

confidence: 99%

“…Given τ -structure G and H, it is useful to study properties in Hom(G, H) that allow us to glue together partially defined homomorphisms. This kind of properties are usually referred in the literature as mixing properties (e.g., see [8,10]).…”

Section: Mixing Propertiesmentioning

confidence: 99%

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Dismantlability, connectedness, and mixing in relational structures

Briceño

Bulatov

Dalmau

et al. 2021

Journal of Combinatorial Theory, Series B

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The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, statistical physics, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, topological properties of the solution set such as connectedness is related to the hardness of CSPs over random structures. In approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions or the free energy of spin systems. Additionally, in the decision CSPs, structural properties of the relational structures involved -like, for example, dismantlability -and their logical characterizations have been instrumental for determining the complexity and other properties of the problem.In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that the structural property of dismantlability of the target graph, the connectedness of the set of homomorphisms, good mixing properties of the corresponding spin system, and the uniqueness of Gibbs measure are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Briceño. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.

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“…In addition it follows from Proposition 5.1 that the graph for the hard square shift (Figure 1) satisfies the hypothesis. For trees with loops, we refer to [8] (Proposition 8.1 and its corollaries) for related results.…”

Section: Phased Mixing Properties For Four-cycle Hom-free Graphsmentioning

confidence: 99%

Mixing properties for hom-shifts and the distance between walks on associated graphs

Chandgotia

Marcus

2018

Pacific J. Math.

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Let H be a finite connected undirected graph and H 2 walk be the graph of bi-infinite walks on H; two such walks {xi} i∈Z and {yi} i∈Z are said to be adjacent if xi is adjacent to yi for all i ∈ Z. We consider the question: Given a graph H when is the diameter (with respect to the graph metric) of H 2 walk finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of Z d with respect to the standard generators to H) and are the subject of this paper.

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Strong Spatial Mixing in Homomorphism Spaces

Cited by 13 publications

References 25 publications

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Dismantlability, connectedness, and mixing in relational structures

Mixing properties for hom-shifts and the distance between walks on associated graphs

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