Uniqueness and global Markov property for Euclidean fields: The case of general polynomial interactions

We give a general method for proving uniqueness and global Markov property for Euclidean quantum fields. The method is based on uniform continuity of local specifications (proved by using potential theoretical tools) and exploitation of a suitable FKG-order structure. We apply this method to give a proof of uniqueness and global Markov property for the Gibbs states and to study extremality of Gibbs states also in the case of non-uniqueness. In particular we prove extremality for ₂ ⁴ (also in the case of non-uniqueness), and global Markov property for weak coupling ₂ ⁴ (which solves a long-standing problem). Uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field.     

Uniqueness and the global Markov property for Euclidean fields. The case of trigonometric interactions

We prove the global Markov property for the Euclidean measure given by weak trigonometric interactions. To obtain this result we first prove a uniqueness theorem concerning the set of regular Gibbs measures corresponding to a given interaction.     

Uniqueness and the global Markov property for Euclidean fields: The case of general exponential interaction

The uniqueness and the global Markov property for the regular Gibbs measure corresponding to the interaction $$U_\Lambda (\varphi ): = \lambda \int\limits_\Lambda {d_2 x\int {d\varrho (\alpha ):e^{\alpha \varphi } :_0 (x)} } $$ [forλ>0,d?(α) a probability measure with support in \(( - 2\sqrt {\pi ,} 2\sqrt \pi )\) ] is proved.     

Some examples concerning the global Markov property

We present examples of interactions of classical lattice systems whose extremal Gibbs states fail to have the global Markov property. One of the examples is translation invariant.Research supported by Natural Sciences and Engineering Research Council grant A4015     

Remarks on the global Markov property

The global Markov property is established for the + state and the – state of attractive lattice systems (e.g., the ferromagnetic Ising model and most other systems for which the FKG inequalities are satisfied) and of the (continuum) Widom Rowlinson model.Supported in part by NSF Grant No. PHY 78-03816     

Sequential Gibbs Measures and Factor Maps

We define the notion of sequential Gibbs measures, inspired by on the classical notion of Gibbs measures and recent examples from the study of non-uniform hyperbolic dynamics. Extending previous results of Kempton and Pollicott (Factors of Gibbs measures for full shifts, entropy of hidden Markov processes and connections to dynamical systems, Cambridge University Press, Cambridge, 2011) and Chazottes and Ugalde (On the preservation of Gibbsianness under symbol amalgamation, entropy of hidden Markov processes and connections to dynamical systems, Cambridge University Press, Cambridge, 2011), we show that the images of one block factor maps of a sequential Gibbs measure are also a sequential Gibbs measure, with the same sequence of Gibbs times. We obtain some estimates on the regularity of the potential of the image measure at almost every point.     

Compactness and the maximal Gibbs state for random Gibbs fields on a lattice

We prove for a general class of Gibbsian Random Field on that the set of tempered Gibbs states is compact. This class contains the Euclidean random fields. Moreover if the interaction is attractive, there is a unique minimal and maximal Gibbs state _– and ₊×_± are unique translation invariant ant and have the global Markov property. We also prove that uniqueness of the tempered Gibbs state is equivalent to the magnetizationsm _±=_±(q _x ) being equal which is true if the pressure is differentiable.     

On Factor Maps that Send Markov Measures to Gibbs Measures

Let X and Y be mixing shifts of finite type. Let π be a factor map from X to Y that is fiber-mixing, i.e., given \(x,\bar{x}\in X\) with \(\pi(x)=\pi(\bar{x})=y\in Y\), there is z∈π ^?1(y) that is left asymptotic to x and right asymptotic to \(\bar{x}\). We show that any Markov measure on X projects to a Gibbs measure on Y under π (for a Hölder continuous potential). In other words, all hidden Markov chains (i.e. sofic measures) realized by π are Gibbs measures. In 2003, Chazottes and Ugalde gave a sufficient condition for a sofic measure to be a Gibbs measure. Our sufficient condition generalizes their condition and is invariant under conjugacy and time reversal. We provide examples demonstrating our result.     

Discrete lattice systems and the equivalence of microcanonical,canonical and grand canonical Gibbs states

It is proven that a microcanonical Gibbs measure on a classical discrete lattice system is a mixture of canonical Gibbs measures, provided the potential is approximately periodic, has finite range and possesses a commensurability property. No periodicity is imposed on the measure. When the potential is not approximately periodic or does not have the commensurability property, the inclusion does not hold.As a by-product, a new proof is given of the fact that for a large class of potentials, a canonical Gibbs measure is a mixture of grand canonical measures. Thus the equivalence of ensembles is obtained in the sense of identical correlation functions.     

On the Stability of the Quenched State in Mean-Field Spin-Glass Models

While the Gibbs states of spin-glass models have been noted to have an erratic dependence on temperature, one may expect the mean over the disorder to produce a continuously varying quenched state. The assumption of such continuity in temperature implies that in the infinite-volume limit the state is stable under a class of deformations of the Gibbs measure. The condition is satisfied by the Parisi Ansatz, along with an even broader stationarity property. The stability conditions have equivalent expressions as marginal additivity of the quenched free energy. Implications of the continuity assumption include constraints on the overlap distribution, which are expressed as the vanishing of the expectation value for an infinite collection of multi- overlap polynomials. The polynomials can be computed with the aid of a real-replica calculation in which the number of replicas is taken to zero.     

Markov and stability properties of equilibrium states for nearest-neighbor interactions

Consider models on the lattice ^d with finite spin space per lattice point and nearest-neighbor interaction. Under the condition that the transfer matrix is invertible we use a transfer-matrix formalism to show that each Gibbs state is determined by its restriction to any pair of adjacent (hyper)planes. Thus we prove that (also in multiphase regions) translationally invariant states have a global Markov property. The transfer-matrix formalism permits us to view the correlation functions of a classicald-dimensional system as obtained by a linear functional on a noncommutative (quantum) system in (d – 1)-dimensions. More precisely, for reflection positive classical states and an invertible transfer matrix the linear functional is a state. For such states there is a decomposition theory available implying statements on the ergodic decompositions of the classical state ind dimensions. In this way we show stability properties of _ev ^d -ergodic states and the absence of certain types of breaking of translational invariance.     

Non-lacunary Gibbs Measures for Certain Fractal Repellers

In this paper, we study non-uniformly expanding repellers constructed as the limit sets for a non-uniformly expanding dynamical systems. We prove that given a Hölder continuous potential φ satisfying a summability condition, there exists non-lacunary Gibbs measure for φ, with positive Lyapunov exponents and infinitely many hyperbolic times almost everywhere. Moreover, this non-lacunary Gibbs measure is an equilibrium measure for φ.     

Renormalization group at criticality and complete analyticity of constrained models: A numerical study

We study the majority rule transformation applied to the Gibbs measure for the 2D Ising model at the critical point. The aim is to show that the renormalized Hamiltonian is well defined in the sense that the renormalized measure is Gibbsian. We analyze the validity of Dobrushin-Shlosman uniqueness (DSU) finite-size condition for the constrained models corresponding to different configurations of the image system. It is known that DSU implies, in our 2D case, complete analyticity from which, as recently shown by Haller and Kennedy. Gibbsianness follows. We introduce a Monte Carlo algorithm to compute an upper bound to Vasserstein distance (appearing in DSU) between finite-volume Gibbs measures with different boundary conditions. We get strong numerical evidence that indeed the DSU condition is verified for a large enough volumeV for all constrained models.     

Local and global Markoff fields

We discuss the relation between local and global Markoff property for processes and (generalized) random fields, both in the case of discrete and continuous ”time parameters“. In particular, we give a short account of our proof of the global Markoff property for the Euclidean Sine-Gordon fields in two dimensions. We also give an account of our proof of the uniqueness of the solutions of the DLR equations in this case (extremality of the Gibbs state, implying in particular uniqueness of the phase). We also discuss more generally the relation between the uniqueness and global Markoff properties, and discuss some consequences of the global Markoff property. Finally, we indicate shortly how the global Markoff property is proven (by ourselves together with G. Olsen) in the case of lattice systems.     

Entropy and global Markov properties

We extend, refine and give simple proofs of some recent results on the validity of global Markov properties for classical spin systems. One of the new results is that there is a global Markov property that is satisfied by equilibrium states in general. The proof of this establishes formulas for the entropy and free energy that show that these quantities are, ford-dimensional systems, given in terms of (d–1)-dimensional systems. Furthermore, we show that global Markov properties imply the absence of some types of symmetry breaking.Research supported by NSF Grants DMS 85-12505 and SMR 86-12369     

The global Markov property for equilibrium states which are determined by correlations in a strip

We present a necessary and sufficient condition for the non-existence of rotational invariant circles for area-preserving twist maps of the cyclinder or annulus based on the cone-crossing and killends criteria of MacKay and Percival (1985). Given a number of technical restrictions on the implementation of these criteria, this condition leads to a proof of MacKay and Percival's Finite Computation Conjecture.     

On the Uniqueness of Gibbs Measures for <Emphasis Type="Italic">p</Emphasis>-Adic Nonhomogeneous λ-Model on the Cayley Tree

We consider a nearest-neighbor p-adic -model with spin values ±1 on a Cayley tree of order k 1. We prove for the model there is no phase transition and as well as being unique, the p-adic Gibbs measure is bounded if and only if p 3. If p=2, then we find a condition which guarantees the nonexistence of a phase transition. Besides, the results are applied to the p-adic Ising model and we show that for the model there is a unique p-adic Gibbs measure.     

Specifications and Martin boundaries forP(Φ)2-random fields

It is shown thatP()₂-Gibbs states in the sense of Guerra, Rosen and Simon are given by a specification. The construction of the specification is based on finding a proper version of the interaction density given by the polynomialP. The existence of this version follows from the fact that all powers of the solution of a Dirichlet problem for an open bounded setU with boundary data given by a distribution are integrable onU. As a consequence the Martin boundary theory for specifications can be applied toP()₂-random fields. It follows that anyP()₂-Gibbs state can be represented in terms of extreme Gibbs states. In certain cases the extreme Gibbs states are characterized in terms of harmonic functions. It follows, in particular, that for any given boundary condition introduced so far the associated cutoffP()₂-measure has a representation as an integral over harmonic functions.     

Steady-state electrical conduction in the periodic Lorentz gas

We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a thermostat constructed according to Gauss principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohm's transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimensiond=2, the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions.Dedicated to Elliott Lieb