269 research outputs found

    On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

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    For a graph G=(V,E)G=(V,E), kNk\in \mathbb{N}, and a complex number ww the partition function of the univariate Potts model is defined as Z(G;k,w):=ϕ:V[k]uvEϕ(u)=ϕ(v)w, {\bf Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi(u)=\phi(v)}}w, where [k]:={1,,k}[k]:=\{1,\ldots,k\}. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any ΔN\Delta\in \mathbb{N} and any keΔ+1k\geq e\Delta+1, there exists an open set UU in the complex plane that contains the interval [0,1)[0,1) such that Z(G;k,w)0{\bf Z}(G;k,w)\neq 0 for any wUw\in U and any graph GG of maximum degree at most Δ\Delta. (Here ee denotes the base of the natural logarithm.) For small values of Δ\Delta we are able to give better results. As an application of our results we obtain improved bounds on kk for the existence of deterministic approximation algorithms for counting the number of proper kk-colourings of graphs of small maximum degree.Comment: In this version the constant 3.02 has been improved to e(=2.71). As a result the entire paper has undergone some changes to accomodate for this improvement. We note that the proofs have in essence not changed much. 22 pages; 2 figure

    On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

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    For a graph G=(V,E), k∈N, and a complex number w the partition function of the univariate Potts model is defined as Z(G;k,w):=∑ϕ:V→[k]∏uv∈Eϕ(u)=ϕ(v)w, where [k]:={1,…,k}. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any Δ∈N and any k≥eΔ+1, there exists an open set U in the complex plane that contains the interval [0,1) such that Z(G;k,w)≠0 for any w∈U and any graph G of maximum degree at most Δ. (Here e denotes the base of the natural logarithm.) For small values of Δ we are able to give better results. As an application of our results we obtain improved bounds on k for the existence of deterministic approximation algorithms for counting the number of proper k-colourings of graphs of small maximum degree

    Factor models on locally tree-like graphs

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    We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree TT, and study the existence of the free energy density ϕ\phi, the limit of the log-partition function divided by the number of vertices nn as nn tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity ϕ\phi subject to uniqueness of a relevant Gibbs measure for the factor model on TT. By way of example we compute ϕ\phi for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on ϕ\phi. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on TT. In the special case that TT has a Galton-Watson law, this formula coincides with the nonrigorous "Bethe prediction" obtained by statistical physicists using the "replica" or "cavity" methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOP828 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Correlation decay and partition function zeros: Algorithms and phase transitions

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    We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite Δ\Delta-regular tree) implies strong spatial mixing on all graphs of maximum degree Δ\Delta can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing. This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard core lattice gas. It also allows us to prove new results on the location of Lee-Yang zeros of the anti-ferromagnetic Ising model. We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs. In particular, we show that results on strong spatial mixing in the anti-ferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function. This extension allows us to give the first deterministic FPTAS for counting the number of qq-colorings of a graph of maximum degree Δ\Delta provided only that q2Δq\ge 2\Delta. This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. We also give an improved version of this result for triangle-free graphs

    A Little Statistical Mechanics for the Graph Theorist

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    In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches. This paper is an elementary general audience survey, intended to popularize the area and provide an accessible first point of entry for further exploration.Comment: 30 pages, 3 figure

    The Potts model and the independence polynomial:Uniqueness of the Gibbs measure and distributions of complex zeros

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    Part 1 of this dissertation studies the antiferromagnetic Potts model, which originates in statistical physics. In particular the transition from multiple Gibbs measures to a unique Gibbs measure for the antiferromagnetic Potts model on the infinite regular tree is studied. This is called a uniqueness phase transition. A folklore conjecture about the parameter at which the uniqueness phase transition occurs is partly confirmed. The proof uses a geometric condition, which comes from analysing an associated dynamical system.Part 2 of this dissertation concerns zeros of the independence polynomial. The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the independence polynomial is related to phase transitions in terms of the analycity of the free energy and plays an important role in the design of efficient algorithms to approximately compute evaluations of the independence polynomial. Chapter 5 directly relates the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. This is done by moreover relating the set of zeros of the independence polynomial to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios. Chapter 6 studies boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. It is shown that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori
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