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63
Graph homomorphisms and phase transitions
 JOURNAL OF COMBINATORIAL THEORY SERIES B
, 1999
"... We model physical systems with "hard constraints" by the space Hom(G, H) of homomorphisms from a locally finite graph G to a fixed finite constraint graph H. For any assignment * of positive real activities to the nodes of H, there is at least one Gibbs measure on Hom(G; H); when G is infi ..."
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Cited by 52 (4 self)
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We model physical systems with "hard constraints" by the space Hom(G, H) of homomorphisms from a locally finite graph G to a fixed finite constraint graph H. For any assignment * of positive real activities to the nodes of H, there is at least one Gibbs measure on Hom(G; H); when G is infinite, there may be more than one.
When G is a regular tree, the simple, invariant Gibbs measures on Hom(G, H) correspond to nodeweighted branching random walks on H. We show that such walks exist for every H and *, and characterize those H which, by admitting more than one such construction, exhibit phase transition behavior.
Towards a Dichotomy Theorem for the Counting Constraint Satisfaction Problem
, 2006
"... The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number ..."
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Cited by 51 (9 self)
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The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number of assignments of values to variables that satisfy all the constraints. The #CSP provides a general framework for numerous counting combinatorial problems including counting satisfying assignments to a propositional formula, counting graph homomorphisms, graph reliability and many others. This problem can be parametrized by the set of relations that may appear in a constraint. In this paper we start a systematic study of subclasses of the #CSP restricted in this way. The ultimate goal of this investigation is to distinguish those restricted subclasses of the #CSP which are solvable in polynomial time from those which are not. We show that the complexity of any restricted #CSP class on a finite domain can be deduced from the properties of polymorphisms of the allowed constraints, similar to that for the decision constraint satisfaction problem. Then we prove that if a subclass of the #CSP is solvable in polynomial time, then constraints allowed by the class satisfy some very restrictive condition: they need to have a Mal’tsev polymorphism, that is a ternary operation m(x, y, z) such that m(x, y, y) = m(y, y, x) = x. This condition uniformly explains many existing complexity results for particular cases of the #CSP, including the dichotomy results for the problem of counting graph homomorphisms, and it allows us to obtain new results.
The complexity of the counting constraint satisfaction problem
 In ICALP (1
, 2008
"... The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structureH can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G toH. In this paper we characterize relational structuresH for which#CSP(H) can ..."
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Cited by 45 (7 self)
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The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structureH can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G toH. In this paper we characterize relational structuresH for which#CSP(H) can be solved in polynomial time and prove that for all other structures the problem is #Pcomplete. 1
A characterization of the entropies of multidimensional shifts of finite type
 Annals of Mathematics
"... Abstract. We show that the values of entropies of multidimensional shifts of finite type (SFTs) are characterized by a certain computationtheoretic property: a real number h≥0is the entropy of such an SFT if and only if it is right recursively enumerable, i.e. there is a computable sequence of rati ..."
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Cited by 36 (7 self)
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Abstract. We show that the values of entropies of multidimensional shifts of finite type (SFTs) are characterized by a certain computationtheoretic property: a real number h≥0is the entropy of such an SFT if and only if it is right recursively enumerable, i.e. there is a computable sequence of rational numbers converging to h from above. The same characterization holds for the entropies of sofic shifts. On the other hand, the entropy of an irreducible SFT is computable. 1.
Equilibrium Measures for Coupled Map Lattices: Existence, Uniqueness and FiniteDimensional Approximations
, 1997
"... Abstract We consider coupled map lattices of hyperbolic type, i.e., chains of weakly interacting hyperbolic sets (attractors) over multidimensional lattices. We describe thermodynamic formalism of the underlying spin lattice system and then prove existence, uniqueness, mixing properties, and expone ..."
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Cited by 34 (3 self)
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Abstract We consider coupled map lattices of hyperbolic type, i.e., chains of weakly interacting hyperbolic sets (attractors) over multidimensional lattices. We describe thermodynamic formalism of the underlying spin lattice system and then prove existence, uniqueness, mixing properties, and exponential decay of correlations of equilibrium measures for a class of Holder continuous potential functions with su ciently small Holder constant. We also study nitedimensional approximations of equilibrium measures in terms of lattice systems (Zapproximations) and lattice spin systems (Z dapproximations). We apply our results to establish existence, uniqueness, and mixing property of SRBmeasures as well as obtain the entropy formula.
Markov Random Fields and Percolation on General Graphs
 Adv. Appl. Probab
, 1999
"... Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation, the Ising model, the WidomRowlinson model and the beach model. Some, but not all, of these implicatio ..."
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Cited by 27 (4 self)
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Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation, the Ising model, the WidomRowlinson model and the beach model. Some, but not all, of these implications hold without the bounded degree assumption. We finally give two examples of (random) unbounded degree graphs in which phase transition in all five models can be established: supercritical GaltonWatson trees, and PoissonVoronoi tessellations of R d for d 2. Keywords: Percolation, Ising model, WidomRowlinson model, beach model, GaltonWatson tree, PoissonVoronoi tessellation. AMS Subject Classification: Primary 60K35, Secondary 82B20, 82B43 1 Introduction Over the last few decades, it has become increasingly clear that there are important connections between percolation theory on one hand, and the issue of Gibbs state multiplicity in Markov random fields on the other. Example...
On the Existence and NonExistence of Finitary Codings for a Class of Random Fields
, 1999
"... We study the existence of finitary codings (also called finitary homomorphisms or finitary factor maps) from a finitevalued i.i.d. process to certain random fields. For Markov random fields we show, using ideas of Marton and Shields, that the presence of a phase transition is an obstruction for the ..."
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Cited by 25 (5 self)
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We study the existence of finitary codings (also called finitary homomorphisms or finitary factor maps) from a finitevalued i.i.d. process to certain random fields. For Markov random fields we show, using ideas of Marton and Shields, that the presence of a phase transition is an obstruction for the existence of the above coding: this yields a large class of Bernoulli shifts for which no such coding exists. Conversely, we show that for the stationary distribution of a monotone exponentially ergodic probabilistic cellular automaton such a coding does exist. The construction of the coding is partially inspired by the ProppWilson algorithm for exact simulation. In particular, combining our results with a theorem of Martinelli and Olivieri, we obtain the fact that for the plus state for the ferromagnetic Ising model on Z d , d 2, there is (not) such a coding when the interaction parameter is below (above) its critical value. Part of this research done while participating, with financ...
The Complexity of Counting Graph Homomorphisms
 In 11th ACM/SIAM Symposium on Discrete Algorithms
, 1999
"... The problem of counting graph homomorphisms is considered. We show that the counting problem corresponding to a given graph is #Pcomplete unless every connected component of the graph is an isolated vertex without a loop, a complete graph with all loops present, or a complete unlooped bipartite gra ..."
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Cited by 23 (5 self)
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The problem of counting graph homomorphisms is considered. We show that the counting problem corresponding to a given graph is #Pcomplete unless every connected component of the graph is an isolated vertex without a loop, a complete graph with all loops present, or a complete unlooped bipartite graph. 1 Introduction Many combinatorial counting problems on graphs can be restated as the problem of counting the number of homomorphisms to a particular graph H. The vertices of H correspond to colours, and the edges show which colours may be adjacent. The graph H may contain loops. Specifically, let C be a set of k colours, where k is a constant. Let H = (C; EH ) be a graph with vertex set C. Given a graph G = (V; E) with vertex set V , a map X : V 7! C is called a Hcolouring if fX(v); X(w)g 2 EH for all fv; wg 2 E: In other words, X is a homomorphism from G to H. Let\Omega H (G) denote the set of all Hcolourings of G. Two wellknown combinatorial counting problems which can be c...
Improved BitStuffing Bounds on TwoDimensional Constraints
 IEEE TRANS. INFORM. THEORY
, 2004
"... We derive lower bounds on the capacity of certain twodimensional (2D) constraints by considering bounds on the entropy of measures induced by bitstuffing encoders. A more detailed analysis of a previously proposed bitstuffing encoder for ()runlengthlimited (RLL) constraints on the square latt ..."
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Cited by 17 (6 self)
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We derive lower bounds on the capacity of certain twodimensional (2D) constraints by considering bounds on the entropy of measures induced by bitstuffing encoders. A more detailed analysis of a previously proposed bitstuffing encoder for ()runlengthlimited (RLL) constraints on the square lattice yields improved lower bounds on the capacity for all 2. This encoding approach is extended to ()RLL constraints on the hexagonal lattice, and a similar analysis yields lower bounds on the capacity for 2. For the hexagonal (1)RLL constraint, the exact coding ratio of the bitstuffing encoder is calculated and is shown to be within 0.5 % of the (known) capacity. Finally, a lower bound is presented on the coding ratio of a bitstuffing encoder for the constraint on the square lattice where each bit is equal to at least one of its four closest neighbors, thereby providing a lower bound on the capacity of this constraint.
Efficient Coding Schemes for the HardSquare Model
, 1999
"... The hardsquare model, also known as the twodimensional (1; 1)RLL constraint, consists of all binary arrays in which the 1's are isolated both horizontally and vertically. Based on a certain probability measure defined on those arrays, an efficient variabletofixed encoder scheme is presente ..."
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Cited by 15 (6 self)
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The hardsquare model, also known as the twodimensional (1; 1)RLL constraint, consists of all binary arrays in which the 1's are isolated both horizontally and vertically. Based on a certain probability measure defined on those arrays, an efficient variabletofixed encoder scheme is presented that maps unconstrained binary words into arrays that satisfy the hardsquare model. For sufficiently large arrays, the average rate of the encoder approaches a value which is only 0.1% below the capacity of the constraint. A second, fixedrate encoder is presented whose rate for large arrays is within 1.2% of the capacity value.