Infinite-volume Gibbs Measure Research Articles

The Blume–Emery–Griffiths Model on the FAD Point and on the AD Line

We analyse the Blume–Emery–Griffiths (BEG) model on the lattice Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^d$$\\end{document} on the ferromagnetic-antiquadrupolar-disordered (FAD) point and on the antiquadrupolar-disordered (AD) line. In our analysis on the FAD point, we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} and a vanishing value in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}. Second, using a careful coupling with the Bernoulli site percolation model in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, we prove rigorously that under imposing +\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+$$\\end{document} boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that there exists a unique zero-temperature infinite-volume Gibbs measure for the BEG. In our analysis of the AD line we restrict ourselves to d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^2$$\\end{document}, we get a condition for the vanishing of the infinite-volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques.

R-positivity and the existence of zero-temperature limits of Gibbs measures on nearest-neighbor matrices

AbstractWe study the $R_\beta$ -positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures $(\mu_{\beta}(\!\cdot\!))_{\beta \geq 0}$ at inverse temperature $\beta$ associated to a family of nearest-neighbor matrices $(Q_{\beta})_{\beta \geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of $\beta$ , and (ii) the existence of weak limits as $\beta \to \infty$ . Some application examples are revised to put in context the main results of this work.

Phase transitions for countable state 1D SOS model with external field

We consider a 1D solid-on-solid (SOS) model with external field in which the single-spin space is the set of all integers. Then, we construct a Gibbs specification for the model and get a functional equation such that every positive solution defines an infinite volume Gibbs measure. We show that there exist infinitely many Gibbs measures for the 1D SOS model with external field for some values of parameters θ and θ 1 (the last one is responsible to the external field). Moreover, by the main theorem, we conclude that a phase transition occurs for the 1D SOS model with external field and there is no Gibbs measure for the 1D SOS model without an external field.

Analyticity for Classical Gasses via Recursion

We give a new criterion for a classical gas with a repulsive pair potential to exhibit uniqueness of the infinite volume Gibbs measure and analyticity of the pressure. Our improvement on the bound for analyticity is by a factor $$e^2$$ over the classical cluster expansion approach and a factor e over the known limit of cluster expansion convergence. The criterion is based on a contractive property of a recursive computation of the density of a point process. The key ingredients in our proofs include an integral identity for the density of a Gibbs point process and an adaptation of the algorithmic correlation decay method from theoretical computer science. We also deduce from our results an improved bound for analyticity of the pressure as a function of the density.

Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order sqrt{log N}. The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop O(2) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.

Models of gradient type with sub-quadratic actions

We consider models of gradient type, which are the densities of a collection of real-valued random variables ϕ ≔ {ϕx: x ∈ Λ} given by Z−1 exp(−∑j∼kV(ϕj − ϕk)). We focus our study on the case that V(∇ϕ)=[1+(∇ϕ)2]α with 0 < α < 1/2, which is a nonconvex potential. We introduce an auxiliary field tjk for each edge and represent the model as the marginal of a model with log-concave density. Based on this method, we prove that finite moments of the fields [v⋅ϕ]p are bounded uniformly in the volume. This leads to the existence of infinite volume measures. Also, every translation invariant, ergodic infinite volume Gibbs measure for the potential V above scales to a Gaussian free field.

Decay of Correlations and Uniqueness of the Infinite-Volume Gibbs Measure of the Canonical Ensemble of 1d-Lattice Systems

We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show the equivalence of correlations of the grand canonical (gce) and the canonical ensemble (ce). As a corollary we obtain that the correlations of the ce decay exponentially plus a volume correction term. Then, we use the decay of correlation to verify a conjecture that the infinite-volume Gibbs measure of the ce is unique on a one-dimensional lattice. For the equivalence of correlations, we modify a method that was recently used by the authors to show the equivalence of the ce and the gce on the level of thermodynamic functions. In this article we also show that the equivalence of the ce and the gce holds on the level of observables. One should be able to extend the methods and results to graphs with bounded degree as long as the gce has a sufficient strong decay of correlations.

Potts model coupled to random causal triangulations

In this work, we study the annealed Potts model coupled to two-dimensional causal triangulations (CTs). Employing duality of graphs, we prove that in the thermodynamic limit, the Potts model coupled to causal triangulations with parameters β and μ is equivalent to a Potts model coupled to dual causal triangulations at the dual parameters β* = log(1 + q/(eβ − 1)) and μ* = μ − 3/2log(eβ − 1) + log q. This duality relation follows from the Fermi–Kurie representation for the Potts model. Employing our duality relation, we determine a region where the critical curve for the annealed model can be located. We also provide lower and upper bounds for the infinite-volume free energy.

Mixing Length Scales of Low Temperature Spin Plaquettes Models

Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we perform a rigorous study of the thermodynamic properties of two dimensional plaquette models, the square and triangular plaquette models. We prove that for any positive temperature both models have a unique infinite volume Gibbs measure with exponentially decaying correlations. We analyse the scaling of three a priori different static correlation lengths in the small temperature regime, the mixing, cavity and multispin correlation lengths. Finally, using the symmetries of the model we determine an exact self similarity property for the infinite volume Gibbs measure.

Critical region for an Ising model coupled to causal triangulations

This paper extends the results obtained by Hernández et al for the annealed Ising model coupled to two-dimensional causal dynamical triangulations. We employ the Fortuin−Kasteleyn (FK) representation in order to determine a region in the quadrant of the parameters where the critical curve for the annealed model is possibly located. This can be done by outlining a region where the model has a unique infinite-volume Gibbs measure, and a region where the finite-volume Gibbs measure does not have weak limit (in fact, does not exist if the volume is large enough). We also improve the region where the model has a one dimensional geometry with respect to the unique weak limit measure, which implies that the Ising model on causal triangulation does not have phase transition in this region. Furthermore, we provide a better approximation of the free energy for the coupled model.

Second-order Markov random fields for independent sets on the infinite Cayley tree

Recently, there has been significant interest in understanding the properties of Markov random fields (M.r.f.) defined on the independent sets of sparse graphs. When these M.r.f. are restricted to pairwise interactions (i.e., hardcore model), much progress has been made. However, considerably less is known in the presence of higher-order interactions, which arise, for example, in the analysis of independent sets with special properties and the study of resource-constrained communication networks. In this paper, we further our understanding of such models by analyzing M.r.f. with second-order interactions on the independent sets of the infinite Cayley tree. We prove that the associated Gibbsian specification satisfies the celebrated FKG inequality whenever the local potentials defining the Hamiltonian satisfy a log-convexity condition. Under this condition, we give necessary and sufficient conditions for the existence of a unique infinite-volume Gibbs measure in terms of an explicit system of equations, prove the existence of a phase transition and give explicit bounds on the associated critical activity, which we prove to exhibit a certain robustness. For potentials which are small perturbations of those coinciding to the hardcore model at the critical activity, we characterize whether the resulting specification has a unique infinite-volume Gibbs measure in terms of whether these perturbations satisfy an explicit linear inequality. Our analysis reveals an interesting nonmonotonicity with regards to biasing toward excluded nodes with no included neighbors.

Thermodynamics for Spatially Inhomogeneous Magnetization and Young-Gibbs Measures

We derive thermodynamic functionals for spatially inhomogeneous magnetization on a torus in the context of an Ising spin lattice model. We calculate the corresponding free energy and pressure (by applying an appropriate external field using a quadratic Kac potential) and show that they are related via a modified Legendre transform. The local properties of the infinite volume Gibbs measure, related to whether a macroscopic configuration is realized as a homogeneous state or as a mixture of pure states, are also studied by constructing the corresponding Young-Gibbs measures.

The approach of Otto-Reznikoff revisited

In this article we consider a lattice system of unbounded continuous spins. Otto and Reznikoff used the two-scale approach to show that exponential decay of correlations yields a logarithmic Sobolev inequality (LSI) with uniform constant in the system size. We improve their statement by weakening the assumptions, for which a more detailed analysis based on two new ingredients is needed. The two new ingredients are a covariance estimate and a uniform moment estimate. We additionally provide a comparison principle for covariances showing that the correlations of a conditioned Gibbs measure are controlled by the correlations of the original Gibbs measure with ferromagnetic interaction. This comparison principle simplifies the verification of the hypotheses of the main result. As an application of the main result we show how sufficient algebraic decay of correlations yields the uniqueness of the infinite-volume Gibbs measure, generalizing a result of Yoshida from finite-range to infinite-range interaction.

Reconstruction for Colorings on Trees

Consider -colorings of the complete tree of depth and branching factor . If we fix the coloring of the leaves, for what range of is the root uniformly distributed over all colors (in the limit )? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which “freeze” the entire tree when . For , Jonasson proved the root is “unbiased” for any fixed coloring of the leaves, and thus the Gibbs measure is unique. What happens for a typical coloring of the leaves? When the leaves have a nonvanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Nonreconstruction is equivalent to extremality of the free-boundary Gibbs measure. When , it is straightforward to show that reconstruction is possible (and hence the measure is not extremal). We prove that for and , nonreconstruction holds; i.e., the Gibbs measure is extremal. We prove a strong form of extremality: With high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. Closely related results were also proven recently by Sly. The above strong form of extremality implies that a local Markov chain that updates constant sized blocks has inverse linear entropy constant and hence mixing time where is the number of vertices of the tree. Extremality on trees and random graphs has received considerable attention recently since it may have connections to the efficiency of local algorithms.

Formula omitted]-positivity of nearest neighbor matrices and applications to Gibbs states

We revisit the R -positivity of nearest neighbor matrices on Z + and the Gibbs measures on the set of nearest neighbor trajectories on Z + whose Hamiltonians award either visits to sites or visits to edges. We give conditions that guarantee the R -positivity or equivalently the existence of the infinite volume Gibbs measure. Moreover, we supply necessary and sufficient conditions for the geometric ergodicity of the associated Markov chain. In this work we generalize and sharpen results obtained in [7,11].

Localization and Perron–Frobenius Theory for Directed Polymers

We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time. We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman--Kac evolution operator. We show that for the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback attractor. Secondly, we use the potential to define a Gibbs specification on paths for any bounded time interval in the usual way and study the thermodynamic limit and existence and uniqueness of an infinite volume Gibbs measure. Both main results claim that the local structure of interaction leads to a unique macroscopic object for almost every realization of the random potential.

Strong Spatial Mixing and Rapid Mixing with Five Colours for the Kagome Lattice

We consider proper 5-colourings of the kagome lattice. Proper q-colourings correspond to configurations in the zero-temperature q-state anti-ferromagnetic Potts model. Salas and Sokal have given a computer assisted proof of strong spatial mixing on the kagome lattice for q ≥ 6 under any temperature, including zero temperature. It is believed that there is strong spatial mixing for q ≥ 4. Here we give a computer assisted proof of strong spatial mixing for q = 5 and zero temperature. It is commonly known that strong spatial mixing implies that there is a unique infinite-volume Gibbs measure and that the Glauber dynamics is rapidly mixing. We give a proof of rapid mixing of the Glauber dynamics on any finite subset of the vertices of the kagome lattice, provided that the boundary is free (not coloured). The Glauber dynamics is not necessarily irreducible if the boundary is chosen arbitrarily for q = 5 colours. The Glauber dynamics can be used to uniformly sample proper 5-colourings. Thus, a consequence of rapidly mixing Glauber dynamics is that there is fully polynomial randomised approximation scheme for counting the number of proper 5-colourings.

Asymptotic behavior of edge-reinforced random walks

In this article, we study linearly edge-reinforced random walk on general multi-level ladders for large initial edge weights. For infinite ladders, we show that the process can be represented as a random walk in a random environment, given by random weights on the edges. The edge weights decay exponentially in space. The process converges to a stationary process. We provide asymptotic bounds for the range of the random walker up to a given time, showing that it localizes much more than an ordinary random walker. The random environment is described in terms of an infinite-volume Gibbs measure.

Strong Spatial Mixing with Fewer Colors for Lattice Graphs

Recursively-constructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontree-like graphs such as lattices. Using this method, we obtain the following general result. Suppose that $G$ is a triangle-free graph and that for some $\degree \geq 3$, the maximum degree of $G$ is at most $\degree$. We show that the spin system consisting of $q$-colorings of $G$ has strong spatial mixing, provided $q > \alpha \degree-\gamma$, where $\alpha\approx 1.76322$ is the solution to $\alpha^\alpha=e$, and $\gamma = \frac{4\alpha^3-6\alpha^2-3\alpha+4}{2(\alpha^2-1)}\approx 0.47031$. Note that we have no additional lower bound on $q$ or $\degree$. This is important for us because our main objective is to have results which are applicable to the lattices studied in statistical physics, such as the integer lattice $\zset^d$ and the triangular lattice. For these graphs (in fact, for any graph in which the distance-$k$ neighborhood of a vertex grows subexponentially in $k$), strong spatial mixing implies that there is a unique infinite-volume Gibbs measure. That is, there is one macroscopic equilibrium rather than many. Our general result gives, for example, a ``hand proof'' of strong spatial mixing for $7$-colorings of triangle-free $4$-regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp. 551--579] (for the rectangular lattice) and by Bubley, Dyer, Greenhill, and Jerrum [\textit{SIAM J. Comput.}, 29 (1999), pp. 387--400].) It also gives a hand proof of strong spatial mixing for $5$-colorings of triangle-free $3$-regular graphs. (A computer-assisted proof for the special case of the hexagonal lattice was provided earlier by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp. 551--579].) Toward the end of the paper we show how to improve our general technique by considering the geometry of the lattice. The idea is to construct the recursive coupling from a system of recurrences rather than from a single recurrence. We use the geometry of the lattice to derive the system of recurrences. This gives us an analysis with a horizon of more than one level of induction, which leads to improved results. We illustrate this idea by proving strong spatial mixing for $q=10$ on the lattice $\zset^3$. Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us a (machine-assisted) proof of strong spatial mixing for $10$-colorings of the triangular lattice. (Such a proof for $11$ colors was given by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp. 551--579].) For completeness, we also show that our strong spatial mixing proof implies rapid mixing of Glauber dynamics for sampling proper colorings of neighborhood-amenable graphs. (It is known that strong spatial mixing often implies rapid mixing, but existing proofs seem to be written for $\zset^d$.) Thus our strong spatial mixing results give rapid

One-dimensional Random Field Kac's Model: Localization of the Phases

We study the typical profiles of a one dimensional random field Kac model, for values of the temperature and magnitude of the field in the region of two absolute minima for the free energy of the corresponding random field Curie Weiss model. We show that, for a set of realizations of the random field of overwhelming probability, the localization of the two phases corresponding to the previous minima is completely determined. Namely, we are able to construct random intervals tagged with a sign, where typically, with respect to the infinite volume Gibbs measure, the profile is rigid and takes, according to the sign, one of the two values corresponding to the previous minima. Moreover, we characterize the transition from one phase to the other. The analysis extends the one done by Cassandro, Orlandi and Picco in [13].