The topological strong spatial mixing property and new conditions for pressure approximation

In the context of stationary $\mathbb{Z}^d$ nearest-neighbour Gibbs measures $\mu$ satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property (TSSM)) on the support of $\mu$ sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of TSSM for studying strong spatial mixing on systems with hard constraints. We also show that TSSM is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by D. Gamarnik and D. Katz (2009), and B. Marcus and R. Pavlov (2013), who gave a special representation of topological pressure in terms of conditional probabilities.Comment: 40 pages, 8 figures. arXiv admin note: text overlap with arXiv:1309.1873 by other author

Representation and poly-time approximation for pressure of Z2\mathbb{Z}^2 lattice models in the non-uniqueness region

We develop a new pressure representation theorem for nearest-neighbour Gibbs interactions and apply this to obtain the existence of efficient algorithms for approximating the pressure in the $2$-dimensional ferromagnetic Potts, multi-type Widom-Rowlinson and hard-core models. For Potts, our results apply to every inverse temperature but the critical. For Widom-Rowlinson and hard-core, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of our work is in the latter.Comment: 37 pages, 2 figure

Dismantlability, connectedness, and mixing in relational structures

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, 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. Topological properties of the solution set such as connectedness are related to the hardness of CSPs over random structures. Additionally, 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 and free energy. 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 dismantlability of the target graph, connectedness of the set of homomorphisms, and good mixing properties of the corresponding spin system 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 Brice\~no. 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.Comment: 27 pages, full version of the paper accepted to ICALP 201

Ecos de la academia: Revista de la Facultad de Educación, Ciencia y Tecnología - FECYT Nro 6

Ecos de la academia, Revista de la Facultad de Educación Ciencia y Tecnología es una publicación científica de la Universidad Técnica del Norte, con revisión por pares a doble ciego que publica artículos en idioma español, quichua, portugués e inglés. Se edita con una frecuencia semestral con dos números por año.En ella se divulgan trabajos originales e inéditos generados por los investigadores, docentes y estudiantes de la FECYT, y contribuciones de profesionales de instituciones docentes e investigativas dentro y fuera del país, con calidad, originalidad y relevancia en las áreas de ciencias sociales y tecnología aplicada.Modelos multidimensionales del bienestar en contextos de enseñanza- aprendizaje: una revisión sistemática. Nuevas tendencias para el área académica de la Publicidad en la zona 1 del Ecuador. Propuesta de un curso de escritura académica bajo la base de modelos experienciales. Aproximación al estudio de las emociones. Seguimiento a egresados y graduados para actualizar el perfil de egreso y profesional. Impacto de la Gerencia de Calidad en el clima organizacional en Educación Básica. Comunicación efectiva del gerente educativo orientada al manejo de conflictos en el personal docente. Meritocracia: Democratización o exclusión en el acceso a la educación superior en Ecuador. Asertividad y desempeño académico en estudiantes universitarios. La creatividad en la formación profesional. Aspectos metodológicos en el proceso de enseñanza- aprendizaje de la gimnasia en estudiantes de Educación Física. English Language Learning Interaction through Web 2.0 Technologies. La sistematización de la práctica educativa y su relación con la metodología de la investigación. El ozono y la oxigenación hiperbárica: una vía para mejorar la recuperación en lesiones deportivas. La labor tutorial: Independencia del aprendizaje en el contexto universitario. Motivación hacia la profesión docente en la Enseñanza Secundaria. El uso académico de Facebook y WhatsApp en estudiantes universitarios... La educación superior en Ecuador: situación actual y factores de mejora de la calidad. El Proyecto de Investigación “Imbabura Étnica”

Combinatorial aspects of spatial mixing and new conditions for pressure representation

Over the last few decades, there has been a growing interest in a measure-theoretical property of Gibbs distributions known as strong spatial mixing (SSM). SSM has connections with decay of correlations, uniqueness of equilibrium states, approximation algorithms for counting problems, and has been particularly useful for proving special representation formulas and the existence of efficient approximation algorithms for (topological) pressure. We look into conditions for the existence of Gibbs distributions satisfying SSM, with special emphasis in hard constrained models, and apply this for pressure representation and approximation techniques in Z^d lattice models. Given a locally finite countable graph G and a finite graph H, we consider Hom(G,H) the set of graph homomorphisms from G to H, and we study Gibbs measures supported on Hom(G,H). We develop some sufficient and other necessary conditions on Hom(G,H) for the existence of Gibbs specifications satisfying SSM (with exponential decay). In particular, we introduce a new combinatorial condition on the support of Gibbs distributions called topological strong spatial mixing (TSSM). We establish many useful properties of TSSM for studying SSM on systems with hard constraints, and we prove that TSSM combined with SSM is sufficient for having an efficient approximation algorithm for pressure. We also show that TSSM is, in fact, necessary for SSM to hold at high decay rate. Later, we prove a new pressure representation theorem for nearest-neighbour Gibbs interactions on Z^d shift spaces, and apply this to obtain efficient approximation algorithms for pressure in the Z² (ferromagnetic) Potts, (multi-type) Widom-Rowlinson, and hard-core lattice gas models. For Potts, the results apply to every inverse temperature except the critical. For Widom-Rowlinson and hard-core lattice gas, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of this work is in the latter, where SSM cannot hold.Science, Faculty ofMathematics, Department ofGraduat

The structure of communication problems in cellular automata

Studying cellular automata with methods from communication complexity appears to be a promising approach. In the past, interesting connections between communication complexity and intrinsic universality in cellular automata were shown. One of the last extensions of this theory was its generalization to various “communication problems”, or “questions ” one might ask about the dynamics of cellular automata. In this article, we aim at structuring these problems, and find what makes them interesting for the study of intrinsic universality and quasi-orders induced by simulation relations