23 research outputs found
Phase coexistence and torpid mixing in the 3-coloring model on Z^d
We show that for all sufficiently large d, the uniform proper 3-coloring model (in physics called the 3-state antiferromagnetic Potts model at zero temperature) on Z^d admits multiple maximal-entropy Gibbs measures. This is a consequence of the following combinatorial result: if a proper 3-coloring is chosen uniformly from a box in Z^d, conditioned on color 0 being given to all the vertices on the boundary of the box which are at an odd distance from a fixed vertex v in the box, then the probability that v gets color 0 is exponentially small in d. The proof proceeds through an analysis of a certain type of cutset separating v from the boundary of the box, and builds on techniques developed by Galvin and Kahn in their proof of phase transition in the hard-core model on Z^d. Building further on these techniques, we study local Markov chains for sampling proper 3-colorings of the discrete torus Z^d_n. We show that there is a constant \rho \approx 0.22 such that for all even n \geq 4 and d sufficiently large, if M is a Markov chain on the set of proper 3-colorings of Z^d_n that updates the color of at most \rho n^d vertices at each step and whose stationary distribution is uniform, then the mixing time of M (the time taken for M to reach a distribution that is close to uniform, starting from an arbitrary coloring) is essentially exponential in n^{d-1}
Torpid Mixing of Markov Chains for the Six-vertex Model on Z^2
In this paper, we study the mixing time of two widely used Markov chain algorithms for the six-vertex model, Glauber dynamics and the directed-loop algorithm, on the square lattice Z^2. We prove, for the first time that, on finite regions of the square lattice these Markov chains are torpidly mixing under parameter settings in the ferroelectric phase and the anti-ferroelectric phase
H-coloring Tori
For graphs G and H, an H-coloring of G is a function from the vertices of G to the vertices of H that preserves adjacency. H-colorings encode graph theory notions such as independent sets and proper colorings, and are a natural setting for the study of hard-constraint models in statistical physics. We study the set of H-colorings of the even discrete torus View the MathML source, the graph on vertex set {0,âŚ,mâ1}d (m even) with two strings adjacent if they differ by 1 (mod m) on one coordinate and agree on all others. This is a bipartite graph, with bipartition classes E and O. In the case m=2 the even discrete torus is the discrete hypercube or Hamming cube Qd, the usual nearest neighbor graph on {0,1}d. We obtain, for any H and fixed m, a structural characterization of the space of H-colorings of View the MathML source. We show that it may be partitioned into an exceptional subset of negligible size (as d grows) and a collection of subsets indexed by certain pairs (A,B)âV(H)2, with each H-coloring in the subset indexed by (A,B) having all but a vanishing proportion of vertices from E mapped to vertices from A, and all but a vanishing proportion of vertices from O mapped to vertices from B. This implies a long-range correlation phenomenon for uniformly chosen H-colorings of View the MathML source with m fixed and d growing. The special pairs (A,B)âV(H)2 are characterized by every vertex in A being adjacent to every vertex in B, and having |A||B| maximal subject to this condition. Our main technical result is an upper bound on the probability, for an arbitrary edge uv of View the MathML source, that in a uniformly chosen H-coloring f of View the MathML source the pair View the MathML source is not one of these special pairs (where Nâ
indicates neighborhood). Our proof proceeds through an analysis of the entropy of f, and extends an approach of Kahn, who had considered the case of m=2 and H a doubly infinite path. All our results generalize to a natural weighted model of H-colorings
Tunneling behavior of Ising and Potts models in the low-temperature regime
We consider the ferromagnetic -state Potts model with zero external field
in a finite volume and assume that the stochastic evolution of this system is
described by a Glauber-type dynamics parametrized by the inverse temperature
. Our analysis concerns the low-temperature regime ,
in which this multi-spin system has stable equilibria, corresponding to the
configurations where all spins are equal. Focusing on grid graphs with various
boundary conditions, we study the tunneling phenomena of the -state Potts
model. More specifically, we describe the asymptotic behavior of the first
hitting times between stable equilibria as in probability,
in expectation, and in distribution and obtain tight bounds on the mixing time
as side-result. In the special case , our results characterize the
tunneling behavior of the Ising model on grid graphs.Comment: 13 figure
Spatial Mixing and Non-local Markov chains
We consider spin systems with nearest-neighbor interactions on an -vertex
-dimensional cube of the integer lattice graph . We study the
effects that exponential decay with distance of spin correlations, specifically
the strong spatial mixing condition (SSM), has on the rate of convergence to
equilibrium distribution of non-local Markov chains. We prove that SSM implies
mixing of a block dynamics whose steps can be implemented
efficiently. We then develop a methodology, consisting of several new
comparison inequalities concerning various block dynamics, that allow us to
extend this result to other non-local dynamics. As a first application of our
method we prove that, if SSM holds, then the relaxation time (i.e., the inverse
spectral gap) of general block dynamics is , where is the number of
blocks. A second application of our technology concerns the Swendsen-Wang
dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies
an bound for the relaxation time. As a by-product of this implication we
observe that the relaxation time of the Swendsen-Wang dynamics in square boxes
of is throughout the subcritical regime of the -state
Potts model, for all . We also prove that for monotone spin systems
SSM implies that the mixing time of systematic scan dynamics is . Systematic scan dynamics are widely employed in practice but have
proved hard to analyze. Our proofs use a variety of techniques for the analysis
of Markov chains including coupling, functional analysis and linear algebra
Algorithmic Pirogov-Sinai theory
We develop an efficient algorithmic approach for approximate counting and
sampling in the low-temperature regime of a broad class of statistical physics
models on finite subsets of the lattice and on the torus
. Our approach is based on combining contour
representations from Pirogov-Sinai theory with Barvinok's approach to
approximate counting using truncated Taylor series. Some consequences of our
main results include an FPTAS for approximating the partition function of the
hard-core model at sufficiently high fugacity on subsets of with
appropriate boundary conditions and an efficient sampling algorithm for the
ferromagnetic Potts model on the discrete torus at
sufficiently low temperature