Infinite-volume Gibbs Measures Research Articles

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.

Convergence to the thermodynamic limit for random-field random surfaces

We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient. In this work we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions $d\geq4$ while the distribution of the surface itself converges in dimensions $d\geq 5$. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by Cotar and Kulske

Long-range order in the 3-state antiferromagnetic Potts model in high dimensions

We prove the existence of long-range order for the 3-state Potts antiferromagnet at low temperature on \mathbb Z^d for sufficiently large d . In particular, we show the existence of six extremal and ergodic infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one bipartition class have a much higher probability to be in one state than in either of the other two states. This settles the high-dimensional case of the Kotecký conjecture.

A $(2+1)$-dimensional growth process with explicit stationary measures

We introduce a class of (2+1)-dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. "Irreversible" means that the interface has an average non-zero drift. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. The model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting Hammersley processes. When the dynamical asymmetry parameter $(p-q)$ equals zero, the infinite-volume Gibbs measures $\pi_\rho$ (with given slope $\rho$) are stationary and reversible. When $p\ne q$, $\pi_\rho$ are not reversible any more but, remarkably, they are still stationary. In such stationary states, we find that the average height function at any given point $x$ grows linearly with time $t$ with a non-zero speed: $\mathbb E Q_x(t):=\mathbb E(h_x(t)-h_x(0))= V(\rho) t$ while the typical fluctuations of $Q_x(t)$ are smaller than any power of $t$ as $t\to\infty$. In the totally asymmetric case of $p=0,q=1$ and on the hexagonal lattice, the dynamics coincides with the "anisotropic KPZ growth model" introduced by A. Borodin and P. L. Ferrari. For a suitably chosen, "integrable", initial condition (that is very far from the stationary state), they were able to determine the hydrodynamic limit and a CLT for interface fluctuations on scale $\sqrt{\log t}$, exploiting the fact that in that case certain space-time height correlations can be computed exactly.

Continuum percolation for Gibbsian point processes with attractive interactions

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large β). The main results are bounds on percolation thresholds ρ ± (β) in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster size distributions. This formula establishes a link with the Gibbs variational principle and a form of equivalence of ensembles, and allows us to combine knowledge on finite volume, canonical Gibbs measures with infinite volume, grand-canonical Gibbs measures

Gibbs measures on permutations over one-dimensional discrete point sets

We consider Gibbs distributions on permutations of a locally finite infinite set $X\subset\mathbb{R}$, where a permutation $\sigma$ of $X$ is assigned (formal) energy $\sum_{x\in X}V(\sigma(x)-x)$. This is motivated by Feynman's path representation of the quantum Bose gas; the choice $X:=\mathbb{Z}$ and $V(x):=\alpha x^2$ is of principal interest. Under suitable regularity conditions on the set $X$ and the potential $V$, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.

Entropy-Driven Phase Transition in Low-Temperature Antiferromagnetic Potts Models

We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane quadrangulations, including the diced lattice. More precisely, we show the existence of (at least) three infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one sublattice have a higher probability to be in one state than in either of the other two states. For the special case of the diced lattice, we give a good rigorous lower bound on this probability, based on computer-assisted calculations that are not available for the other lattices.

A finite-volume version of Aizenman–Higuchi theorem for the 2d Ising model

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model are convex combinations of the two pure phases. We present here a new approach to this result, with a number of advantages: (i) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (ii) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (iii) this new approach might be applicable to systems for which the classical method fails.

Spontaneous breaking of continuous rotational symmetry in two dimensions

In this article, we consider a simple model in equilibrium statistical mechanics for a two-dimensional crystal without defects. In this model, the local specifications for infinite-volume Gibbs measures are rotationally symmetric. We show that at sufficiently low, but positive temperature, rotational symmetry is spontaneously broken in some infinite-volume Gibbs measures.

A remark on the infinite-volume Gibbs measures of spin glasses

In this note, we point out that infinite-volume Gibbs measures of spin glass models on the hypercube can be identified as random probability measures on the unit ball of a Hilbert space. This simple observation follows from a result of Dovbysh and Sudakov on weakly exchangeable random matrices. Limiting Gibbs measures can then be studied as single well-defined objects. This approach naturally extends the space of random overlap structures as defined by Aizenman et al. We discuss the Ruelle probability cascades and the stochastic stability within this framework. As an application, we use an idea of Parisi and Talagrand to prove that if a sequence of finite-volume Gibbs measures satisfies the Ghirlanda–Guerra identities, then the infinite-volume measure must be singular as a measure on a Hilbert space.

Phase coexistence of gradient Gibbs states

We consider the (scalar) gradient fields η a = (η b )—with b denoting the nearest-neighbor edges in \(\mathbb{Z}^{2}\)—that are distributed according to the Gibbs measure proportional to \(\hbox{e}^{-\beta H(\eta)}\nu(d \eta)\). Here H = ∑ b V(η b ) is the Hamiltonian, V is a symmetric potential, β > 0 is the inverse temperature, and ν is the Lebesgue measure on the linear space defined by imposing the loop condition \(\eta_{b_1}+\eta_{b_2}=\eta_{b_3}+\eta_{b_4}\) for each plaquette (b 1,b 2,b 3,b 4) in \(\mathbb{Z}^{2}\). For convex V, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their tilt. We describe a mechanism by which the gradient Gibbs measures with non-convex V undergo a structural, order-disorder phase transition at some intermediate value of inverse temperature β. At the transition point, there are at least two distinct gradient measures with zero tilt, i.e., E η b = 0.

Random dynamics and thermodynamic limits for polygonal Markov fields in the plane

We construct random dynamics for collections of nonintersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1983) and Arak and Surgailis (1989), (1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernándezet al.(1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.

Random dynamics and thermodynamic limits for polygonal Markov fields in the plane

We construct random dynamics for collections of nonintersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1983) and Arak and Surgailis (1989), (1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernández et al. (1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.

DIFFUSION PROCESSES ON PATH SPACES WITH INTERACTIONS

We construct dynamics on path spaces C (ℝ; ℝ) and C([-r, r];ℝ) whose equilibrium states are Gibbs measures with free potential φ and interaction potential ψ. We do this by using the Dirichlet form theory under very mild conditions on the regularity of potentials. We take the carré du champ similar to the one of the Ornstein–Uhlenbeck process on C([0, ∞);ℝ). Our dynamics are non-Gaussian because we take Gibbs measures as reference measures. Typical examples of free potentials are double-well potentials and interaction potentials are convex functions. In this case the associated infinite-volume Gibbs measures are singular to any Gaussian measures on C(ℝ;ℝ).

Gibbs states of the Hopfield model in the regime of perfect memory

We study the thermodynamic properties of the Hopfield model of an autoassociative memory. IfN denotes the number of neurons andM (N) the number of stored patterns, we prove the following results: IfM/N↓ 0 asN↑ ∞, then there exists an infinite number of infinite volume Gibbs measures for all temperaturesT<1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point and the Gibbs measures on spin configurations are products of Bernoulli measures. IfM/N → α, asN↓∞ for α small enough, we show that for temperaturesT smaller than someT(α)<1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

For Gibbsian systems of particles inR d , we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.