Abstract:
We propose a hierarchy for approximate inference based on the Dobrushin, Lanford, Ruelle (DLR) equations. This hierarchy includes existing algorithms, such as belief propagation, and also motivates novel algorithms such as factorized neighbors (FN) algorithms and variants of mean field (MF) algorithms. In particular, we show that extrema of the Bethe free energy correspond to approximate solutions of the DLR equations. In addition, we demonstrate a close connection between these approximate algorithms and Gibbs sampling. Finally, we compare and contrast various of the algorithms in the DLR hierarchy on spin-glass problems. The experiments show that algorithms higher up in the hierarchy give more accurate results when they converge but tend to be less stable.

Abstract:
The one-dimensional deterministic economic model recently studied by Gonzalez-Estevez et al. [Physica A 387, 4367 (2008)] is considered on a two-dimensional square lattice with periodic boundary conditions. In this model, the evolution of each agent is described by a map coupled with its nearest neighbors. The map has two factors: a linear term that accounts for the agent's own tendency to grow and an exponential term that saturates this growth through the control effect of the environment. The regions in the parameter space where the system displays Pareto and Boltzmann-Gibbs statistics are calculated for the cases of von Neumann and of Moore's neighborhoods. It is found that, even when the parameters in the system are kept fixed, a transition from Pareto to Boltzmann-Gibbs behavior can occur when the number of neighbors of each agent increases.

Abstract:
The paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in a Euclidean space. The kinetic energy part of the Hamiltonian is the standard Laplacian (with a Dirichlet's boundary condition at the border of a `box'). The particles interact with each other through a two-body finite-range potential depending on the distance between them and featuring a hard core of a positive diameter. We introduce a class of so-called FK-DLR functionals containing all limiting Gibbs states of the system. In the next paper we will prove that any FK-DLR functional is shift-invariant, regardless of whether it is unique or not.

Abstract:
In a previous paper by the authors the existence of Haar projections with growing norms in Sobolev-Triebel-Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel-Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.

Abstract:
We give new and explicitly computable examples of Gibbs-non-Gibbs transitions of mean-field type, using the large deviation approach introduced in [4]. These examples include Brownian motion with small variance and related diffusion processes, such as the Ornstein-Uhlenbeck process, as well as birth and death processes. We show for a large class of initial measures and diffusive dynamics both short-time conservation of Gibbsianness and dynamical Gibbs-non-Gibbs transitions.

Abstract:
The Gibbs states of a spin system on the lattice Zd with pair interactions Jxyσ(x) σ(y) are studied. Here ∈ E, i.e. x and y are neighbors in Zd. The intensities Jxy and the spins σ(x), σ(y) are arbitrarily real. To control their growth we introduce appropriate sets Jq RE and Sp RZd and show that, for every J = (Jxy)∈Jq: (a) the set of Gibbs states Gp(J) = {μ: solves DLR, μ(Sp) = 1} is non-void and weakly compact; (b) each μ∈Gp(J) obeys an integrability estimate, the same for all μ. Next we study the case where Jq is equipped with a norm, with the Borel σ-field B(Jq), and with a complete probability measure ν. We show that the set-valued map Jq J → Gp(J) has measurable selections Jq J → μ(J) ∈Gp(J), which are random Gibbs measures. We demonstrate that the empirical distributions N-1Σn=1NπΔn(·|J,ξ), obtained from the local conditional Gibbs measures πΔn(·|J,ξ) and from exhausting sequences of Δn Zd, have ν-a.s. weak limits as N→+∞, which are random Gibbs measures. Similarly, we show the existence of the ν-a.s. weak limits of the empirical metastates N-1Σn=1NδπΔn(·|J,ξ), which are Aizenman-Wehr metastates. Finally, we demonstrate that the limiting thermodynamic pressure exists under some further conditions on ν.

Abstract:
The Gibbs measures of a spin system on $Z^d$ with unbounded pair interactions $J_{xy} \sigma (x) \sigma (y)$ are studied. Here $\langle x, y \rangle \in E $, i.e. $x$ and $y$ are neighbors in $Z^d$. The intensities $J_{xy}$ and the spins $\sigma (x) , \sigma (y)$ are arbitrary real. To control their growth we introduce appropriate sets $J_q\subset R^E$ and $S_p\subset R^{Z^d}$ and prove that for every $J = (J_{xy}) \in J_q$: (a) the set of Gibbs measures $G_p(J)= \{\mu: solves DLR, \mu(S_p)=1\}$ is non-void and weakly compact; (b) each $\mu\inG_p(J)$ obeys an integrability estimate, the same for all $\mu$. Next we study the case where $J_q$ is equipped with a norm, with the Borel $\sigma$-field $B(J_q)$, and with a complete probability measure $\nu$. We show that the set-valued map $J \mapsto G_p(J)$ is measurable and hence there exist measurable selections $J_q \ni J \mapsto \mu(J) \in G_p(J)$, which are random Gibbs measures. We prove that the empirical distributions $N^{-1} \sum_{n=1}^N \pi_{\Delta_n} (\cdot| J, \xi)$, obtained from the local conditional Gibbs measures $\pi_{\Delta_n} (\cdot| J, \xi)$ and from exhausting sequences of $\Delta_n \subset Z^d$, have $\nu$-a.s. weak limits as $N\rightarrow +\infty$, which are random Gibbs measures. Similarly, we prove the existence of the $\nu$-a.s. weak limits of the empirical metastates $N^{-1} \sum_{n=1}^N \delta_{\pi_{\Delta_n} (\cdot| J,\xi)}$, which are Aizenman-Wehr metastates. Finally, we prove the existence of the limiting thermodynamic pressure under some further conditions on $\nu$.

Abstract:
A deterministic system of interacting agents is considered as a model for economic dynamics. The dynamics of the system is described by a coupled map lattice with near neighbor interactions. The evolution of each agent results from the competition between two factors: the agent's own tendency to grow and the environmental influence that moderates this growth. Depending on the values of the parameters that control these factors, the system can display Pareto or Boltzmann-Gibbs statistical behaviors in its asymptotic dynamical regime. The regions where these behaviors appear are calculated on the space of parameters of the system. Other statistical properties, such as the mean wealth, the standard deviation, and the Gini coefficient characterizing the degree of equity in the wealth distribution are also calculated on the space of parameters of the system.

Abstract:
In this paper we consider deterministic limits of molecular stochastic systems with finite and infinite degrees of freedom. The method to obtain the deterministic vector field is based on the continuum limit of such microscopic systems which has been derived in [11]. With the aid of the theory we finally develop a new approach for molecular systems that describe typical enzyme kinetics or other interactions between molecular machines like genetic elements and smaller communicating molecules. In contrast to the literature on enzyme kinetics the resulting deterministic functional responses are not derived by time-scale arguments on the macroscopic level, but are a result of time scaling transition rates on the discrete microscopic level. We present several examples of common functional responses found in the literature, like Michaelis-Menten and Hill equation. We finally give examples of more complex but typical macro-molecular machinery.

Abstract:
In this paper we consider general continuous potentials and potentials with more regularity on the symbolic space and show how the Dobrushin-Lanford-Ruelle Gibbs measures in the one-dimensional lattice are related to the Gibbs Measure associated to the Ruelle operator. In particular, we show how to obtain the Gibbs Measures for H\"older and Walters potentials considered in Thermodynamic Formalism (for the full shift) via Thermodynamic Limit. An absolutely uniformly summable interaction is associated to every H\"older potential and we show how to construct a DLR specification for any continuous potential. We also show that the Long-Range Ising Model with interactions of the form $1/r^{\alpha}$ for $\alpha>2$ is in the Walter class. We compare Thermodynamic Limit probabilities, DLR probabilities and, motivated by the achieved equivalences, consider the natural concept in Thermodynamic Formalism of dual Gibbs measures. The discussion is conducted from a Ruelle operator perspective, but it should be noted that our results are for the lattice $\mathbb{N}$ and not the usual lattice $\mathbb{Z}$. Employing the newfound connection, we prove some uniform convergence theorems for finite volume Gibbs measures. One of the primary aims of this work is to provide a "dictionary" translating potentials in Thermodynamic Formalism into interactions. Thus one of the contributions is to clarify for both the Dynamical Systems and Mathematical Statisical Mechanics communities how these (seemingly distinct) concepts of Gibbs measures are related.