Hausdorff Dimension of Sets of Generic Points for Gibbs Measures (original) (raw)
Gibbs cluster measures on configuration spaces
2010
The distribution gclg_{cl}gcl of a Gibbs cluster point process in X=mathbbRdX=\mathbb{R}^{d}X=mathbbRd (with i.i.d. random clusters attached to points of a Gibbs configuration with distribution ggg) is studied via the projection of an auxiliary Gibbs measure hatg\hat{g}hatg in the space of configurations hatgamma=(x,bary)subsetXtimesmathfrakX\hat{gamma}=\{(x,\bar{y})\}\subset X\times\mathfrak{X}hatgamma=(x,bary)subsetXtimesmathfrakX, where xinXx\in XxinX indicates a cluster "center" and baryinmathfrakX:=bigsqcupnXn\bar{y}\in\mathfrak{X}:=\bigsqcup_{n} X^nbaryinmathfrakX:=bigsqcupnXn represents a corresponding cluster relative to xxx. We show that the measure gclg_{cl}gcl is quasi-invariant with respect to the group mathrmDiff0(X)\mathrm{Diff}_{0}(X)mathrmDiff0(X) of compactly supported diffeomorphisms of XXX, and prove an integration-by-parts formula for gclg_{cl}gcl. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. These results are quite general; in particular, the uniqueness of the background Gibbs measure ggg is not required. The paper is an extension of the earlier results for Poisson cluster measures %obtained by the authors [J. Funct. Analysis 256 (2009) 432-478], where a different projection construction was utilized specific to this "exactly soluble" case.
Finite type approximations of Gibbs measures on sofic subshifts
Nonlinearity, 2005
Consider a Hölder continuous potential φ defined on the full shift A N , where A is a finite alphabet. Let X ⊂ A N be a specified sofic subshift. It is well-known that there is a unique Gibbs measure µ φ on X associated to φ. Besides, there is a natural nested sequence of subshifts of finite type (Xm) converging to the sofic subshift X. To this sequence we can associate a sequence of Gibbs measures (µ m φ ). In this paper, we prove that these measures weakly converge at exponential speed to µ φ ( in the classical distance metrizing weak topology). We also establish a strong mixing property (ensuring weak Bernoullicity) of µ φ . Finally, we prove that the measure-theoretic entropy of µ m φ converges to the one of µ φ exponentially fast. We indicate how to extend our results to more general subshifts and potentials. We stress that we use basic algebraic tools (contractive properties of iterated matrices) and symbolic dynamics.
Gibbs measures of disordered lattice systems with unbounded spins
2010
The Gibbs measures of a spin system on ZdZ^dZd with unbounded pair interactions Jxysigma(x)sigma(y)J_{xy} \sigma (x) \sigma (y)Jxysigma(x)sigma(y) are studied. Here langlex,yrangleinE\langle x, y \rangle \in E langlex,yrangleinE, i.e. xxx and yyy are neighbors in ZdZ^dZd. The intensities JxyJ_{xy}Jxy and the spins sigma(x),sigma(y)\sigma (x) , \sigma (y)sigma(x),sigma(y) are arbitrary real. To control their growth we introduce appropriate sets JqsubsetREJ_q\subset R^EJqsubsetRE and SpsubsetRZdS_p\subset R^{Z^d}SpsubsetRZd and prove that for every J=(Jxy)inJqJ = (J_{xy}) \in J_qJ=(Jxy)inJq: (a) the set of Gibbs measures Gp(J)=mu:solvesDLR,mu(Sp)=1G_p(J)= \{\mu: solves DLR, \mu(S_p)=1\}Gp(J)=mu:solvesDLR,mu(Sp)=1 is non-void and weakly compact; (b) each muinGp(J)\mu\inG_p(J)muinGp(J) obeys an integrability estimate, the same for all mu\mumu. Next we study the case where JqJ_qJq is equipped with a norm, with the Borel sigma\sigmasigma-field B(Jq)B(J_q)B(Jq), and with a complete probability measure nu\nunu. We show that the set-valued map JmapstoGp(J)J \mapsto G_p(J)JmapstoGp(J) is measurable and hence there exist measurable selections JqniJmapstomu(J)inGp(J)J_q \ni J \mapsto \mu(J) \in G_p(J)JqniJmapstomu(J)inGp(J), which are random Gibbs measures. We prove that the empirical distributions $N^{-1} \sum_...
Equidistribution, ergodicity and irreducibility associated with Gibbs measures
Commentarii Mathematici Helvetici
We generalize an equidistribution theoremà la Bader-Muchnik ([9]) for operatorvalued measures constructed from a family of boundary representations associated with Gibbs measures in the context of convex cocompact discrete group of isometries of a simply connected connected Riemannian manifold with pinched negative curvature. We combine a functional analytic tool, namely the property RD of hyperbolic groups ([32] and [31]), together with a dynamical tool: an equidistribution theorem of Paulin, Pollicott and Schapira inspired by a result of Roblin ([42]). In particular, we deduce irreducibility of these new classes of boundary representations.
Fixed points of an infinite dimensional operator related to Gibbs measures
arXiv (Cornell University), 2022
We describe fixed points of an infinite dimensional non-linear operator related to a hard core (HC) model with a countable set N of spin values on the Cayley tree. This operator is defined by a countable set of parameters λi > 0, aij ∈ {0, 1}, i, j ∈ N. We find a sufficient condition on these parameters under which the operator has unique fixed point. When this condition is not satisfied then we show that the operator may have up to five fixed points. Also, we prove that every fixed point generates a normalisable boundary law and therefore defines a Gibbs measure for the given HC-model.
Global specifications and nonquasilocality of projections of Gibbs measures
1997
We study the question of whether the quasilocality (continuity, almost Markovianness) property of Gibbs measures remains valid under a projection on a sub-σ-algebra. Our method is based on the construction of global specifications, whose projections yield local specifications for the projected measures. For Gibbs measures compatible with monotonicity preserving local specifications, we show that the set of configurations where quasilocality is lost is an event of the tail field. This set is shown to be empty whenever a strong uniqueness property is satisfied, and of measure zero when the original specification admits a single Gibbs measure. Moreover, we provide a criterion for nonquasilocality (based on a quantity related to the surface tension). We apply these results to projections of the extremal measures of the Ising model. In particular, our nonquasilocality criterion allows us to extend and make more complete previous studies of projections to a sublattice of one less dimension (Schonmann example).
Chains with Complete Connections and One-Dimensional Gibbs Measures
Electronic Journal of Probability, 2004
We discuss the relationship between one-dimensional Gibbs measures and discrete-time processes (chains). We consider finite-alphabet (finitespin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specifications). Furthermore, we discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. As an auxiliary result we prove a (re)construction theorem for specifications starting from single-site conditioning, which applies in a more general setting (general spin space, specifications not necessarily Gibbsian).
2005
We identify a sufficiant condition for a sequence of Gibbs measures P λ with the density Z λ e −J 0 (v)−J 1 (v) , v ∈ R n , defined on a state space of v s, to converge weakly in a sense of measures to a Gibbs measure with a density Z λ e −J 0 (v) , where the dominating measure for the density is the Hausdorff measure with an appropriate dimension. The function J 0 identifies an objective and J 1 defines a constraint. The condition we introduce requires the Hessian of J 1 to be non-negative definite and to have a constant rank on each component of {v ∈ R n | J 1 (v) = 0}. The result presented shows that the probability measures P λ concentrate on the highest dimensional stratum of J −1 1 (0). We apply this result to a non-quasiconvex variational problem describing microstructural equilibria of a binary martensitic alloy. We show that the Young's measure describing, in general, non-attainable infimum of such a problem can be obtained as a "push-forward" measure induced by the probablity measure P λ through a linear bounded operator T λ : GM → AY , where GM denotes the space of Gibbs measures, and Y M denotes the space of Young's measures defined as all probability measures generated by gradients of bounded sequences in a suitable Sobolev space. The basic idea of Simulated Annealing (SA) goes back to [8], although it was not given its name until 30 years later [10]. Numerous advances in the general area of "Markov Chain Monte Carlo" (MCMC; see [12]) have lead to extensions of the basic SA algorithm which greatly improve its performance and range of applicability. Suppose our objective is to find arg min v J 0 (v), v ∈ R n. Assume that J 0 is a sufficiently regular function (e.g., continuous), bounded below, and J 0 → ∞ as v → ∞ sufficiently fast that Z −1 = R n e −J0(v) dv < ∞.