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_...

Marked Gibbs measures via cluster expansion

1999

We give a sufficiently detailed account on the construction of marked Gibbs measures in the high temperature and low fugacity regime. This is proved for a wide class of underlying spaces and potentials such that stability and integrability conditions are satisfied. That is, for state space we take a locally compact separable metric space XXX and a separable metric space

Almost Gibbsian versus weakly Gibbsian measures

Stochastic Processes and Their Applications, 1999

We consider various extensions of the standard de nition of Gibbs states for lattice spin systems. When a random eld has conditional distributions which are almost surely continuous (almost Gibbsian eld), then there is a potential for that eld which is almost surely summable (weakly Gibbsian eld). This generalizes the standard Kozlov-Sullivan theorems. The converse is not true in general. We give (counter)examples illustrating the relation between topological and measure-theoretic aspects of generalized Gibbs de nitions.

On the possible failure of the Gibbs property for measures on lattice systems

1996

We review results which have been obtained in the last few years on the topic whether various measures of physical interest, defined on lattice systems, have the Gibbs property or not. Most of the measures we consider occur either within renormalization-group theory or within models of non-equilibrium statistical mechanics. In renormalization-group theory examples we discuss in particular the uniqueness regime and the critical regime. Moreover we consider the question how severely the Gibbs property is violated in various examples.

Attractor properties of non-reversible dynamics w.r.t invariant Gibbs measures on the lattice

arXiv: Probability, 2014

We consider stochastic dynamics of lattice systems with finite local state space, possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: a) There is at least one stationary measure which is a Gibbs measure for an absolutely summable potential �. b) Zero loss of relative entropy density under dynamics implies the Gibbs property with the same �. We prove results on the attractor property of the set of Gibbs measures for �: 1. The set of weak limit points of any trajectory of translation-invariant measures contains at least one Gibbs state for �. 2. We show that if all elements of a weakly convergent sequence of measures are Gibbs measures for a sequence of some translation-invariant summable potentials with uniform bound, then the limiting measure must be a Gibbs measure for �. 3. We give an extension of the second result to trajectories which are allowed to be non-Gibbs, but have a property of asymptotic smallness of ...

Entropy and drift for Gibbs measures on geometrically finite manifolds

Transactions of the American Mathematical Society, 2019

We prove a generalization of the fundamental inequality of Guivarc’h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of C A T ( − 1 ) CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is equivalent to the hitting measure. As a corollary, if the action is not convex cocompact, any hitting measure is singular to any Gibbs density.