Graphical Representations for Ising and Potts Models in General External Fields

This work is concerned with the theory of Graphical Representation for the Ising and Potts Models over general lattices with non-translation invariant external field. We explicitly describe in terms of the Random Cluster Representation the distribution function and, consequently, the expected value of a single spin for the Ising and $q$-states Potts Models with general external fields. We also consider the Gibbs States for the Edwards-Sokal Representation of the Potts Model with non-translation invariant magnetic field and prove a version of the FKG Inequality for the so called General Random Cluster Model (GRC Model) with free and wired boundary conditions in the non-translation invariant case. Adding the amenability hypothesis on the lattice, we obtain the uniqueness of the infinite connected component and the quasilocality of the Gibbs Measures for the GRC Model with such general magnetic fields. As a final application of the theory developed, we show the uniqueness of the Gibbs Measures for the Ferromagnetic Ising Model with a positive power law decay magnetic field, as conjectured in [8].Comment: 56 pages. Accepted for publication in Journal of Statistical Physic

Equivalence of optimal L1L^1-inequalities on Riemannian Manifolds

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \geq 2$. This paper concerns to the validity of the optimal Riemannian $L^1$-Entropy inequality $${\bf Ent}_{dv_g}(u) \leq n \log \left(A_{opt} \|D u\|_{BV(M)} + B_{opt}\right)$$ for all $u \in BV(M)$ with $\|u\|_{L^1(M)} = 1$ and existence of extremal functions. In particular, we prove that this optimal inequality is equivalent a optimal $L^1$-Sobolev inequality obtained by Druet [6].Comment: To appear in Journal of Mathematical Analysis and Its Applications (JMAA

Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach

We consider a family of potentials f, derived from the Hofbauer potentials, on the symbolic space Omega=\{0,1\}^\mathbb{N} and the shift mapping $\sigma$ acting on it. A Ruelle operator framework is employed to show there is a phase transition when the temperature varies in the following senses: the pressure is not analytic, there are multiple eigenprobabilities for the dual of the Ruelle operator, the DLR-Gibbs measure is not unique and finally the Thermodynamic Limit is not unique. Additionally, we explicitly calculate the critical points for these phase transitions. Some examples which are not of Hofbauer type are also considered. The non-uniqueness of the Thermodynamic Limit is proved by considering a version of a Renewal Equation. We also show that the correlations decay polynomially and that each one of these Hofbauer potentials is a fixed point for a certain renormalization transformation.Comment: in Journ. of Stat. Phys 2015; Jour of Stat Phys 201

Phase Transitions in Ferromagnetic Ising Models with spatially dependent magnetic fields

In this paper we study the nearest neighbor Ising model with ferromagnetic interactions in the presence of a space dependent magnetic field which vanishes as $|x|^{-\alpha}$, $\alpha >0$, as $|x|\to \infty$. We prove that in dimensions $d\ge 2$ for all $\beta$ large enough if $\alpha>1$ there is a phase transition while if $\alpha<1$ there is a unique DLR state.Comment: to appear in Communications in Mathematical Physic

Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples

In this paper we construct spectral triples $(A,H,D)$ on the symbolic space when the alphabet is finite. We describe some new results for the associated Dixmier trace representations for Gibbs probabilities (for potentials with less regularity than H\"older) and for a certain class of functions. The Dixmier trace representation can be expressed as the limit of a certain zeta function obtained from high order iterations of the Ruelle operator. Among other things we consider a class of examples where we can exhibit the explicit expression for the zeta function. We are also able to apply our reasoning for some parameters of the Dyson model (a potential on the symbolic space $\{-1,1\}^\mathbb{N}$) and for a certain class of observables. Nice results by R. Sharp, M.~Kesseb\"ohmer and T.~Samuel for Dixmier trace representations of Gibbs probabilities considered the case where the potential is of H\"older class. We also analyze a particular case of a pathological continuous potential where the Dixmier trace representation - via the associated zeta function - is not true.Comment: the tile was modified and there are two more author