269 research outputs found
On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs
For a graph , , and a complex number the
partition function of the univariate Potts model is defined as where . 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 and any
, there exists an open set in the complex plane that
contains the interval such that for any
and any graph of maximum degree at most . (Here denotes the
base of the natural logarithm.) For small values of we are able to
give better results.
As an application of our results we obtain improved bounds on for the
existence of deterministic approximation algorithms for counting the number of
proper -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
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
We consider homogeneous factor models on uniformly sparse graph sequences
converging locally to a (unimodular) random tree , and study the existence
of the free energy density , the limit of the log-partition function
divided by the number of vertices as tends to infinity. We provide a
new interpolation scheme and use it to prove existence of, and to explicitly
compute, the quantity subject to uniqueness of a relevant Gibbs measure
for the factor model on . By way of example we compute 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 . 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 . In the special case that 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
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 -regular tree) implies strong spatial mixing on
all graphs of maximum degree 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
-colorings of a graph of maximum degree provided only that . 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
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
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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