We discuss the relationship between discrete-time processes (chains) and one-dimensional Gibbs measures. We consider finite-alphabet (finite-spin) 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).
In this contribution we discuss the role disordered (or random) systems have played in the study of non-Gibbsian measures. This role has two main aspects, the distinction between which has not always been fully clear: 1) {\em From} disordered systems: Disordered systems can be used as a tool; analogies with, as well as results and methods from the study of random systems can be employed to investigate non-Gibbsian properties of a variety of measures of physical and mathematical interest. 2) {\em Of} disordered systems: Non-Gibbsianness is a property of various (joint) measures describing quenc…
We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature $β^{-1}$ goes to zero. Our main results are: (1) As $β\to \infty$ at fixed positive pressure $p>0$, the Gibbs measures $μ_β$ and $ν_β$ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals $\overline{\mathcal{E}}_{\mathrm{bulk}}$ and $\overline{\mathcal{E}}_\mathr…
We use a non-Markovian coupling and small modifications of techniques from the theory of finite Markov chains to analyze some Markov chains on continuous state spaces. The first is a Gibbs sampler on narrow contingency tables, the second a gen- eralization of a sampler introduced by Randall and Winkler.
We establish a one-to-one correspondence between one-sided and two-sided regular systems of conditional probabilities on the half-line that preserves the associated chains and Gibbs measures. As an application, we determine uniqueness and non-uniqueness regimes in one-sided versions of ferromagnetic Ising models with long range interactions. Our study shows that the interplay between chain and Gibbsian theories yields more information than that contained within the known theory of each separate framework. In particular: (i) A Gibbsian construction due to Dyson yields a new family of chains wit…
A $ϕ$-irreducible and aperiodic Markov chain with stationary probability distribution will converge to its stationary distribution from almost all starting points. The property of Harris recurrence allows us to replace ``almost all'' by ``all,'' which is potentially important when running Markov chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings algorithms are known to be Harris recurrent. In this paper, we consider conditions under which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are not Harris recurrent. We present a simple but natural two-dimensional…
This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper [32], we proved the Eyring-Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of [45], which considered the same problem for metastable reversible diffusion processes. Our proof …
SISSA（国际高等研究学院）；INFN sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；School of Mathematics, University of Bristol（布里斯托大学数学学院）；CNRS, Unité de Mathématiques Pures et Appliquées (UMPA), École Normale Supérieure de Lyon（法国国家科学研究中心，纯粹与应用数学单元（UMPA），里昂高等师范学院）；CNRS, Laboratoire de Physique (LPENS), École Normale Supérieure de Lyon（法国国家科学研究中心，物理实验室（LPENS），里昂高等师范学院）
We establish quantitative bounds for rates of convergence and asymptotic variances for iterated conditional sequential Monte Carlo (i-cSMC) Markov chains and associated particle Gibbs samplers. Our main findings are that the essential boundedness of potential functions associated with the i-cSMC algorithm provide necessary and sufficient conditions for the uniform ergodicity of the i-cSMC Markov chain, as well as quantitative bounds on its (uniformly geometric) rate of convergence. Furthermore, we show that the i-cSMC Markov chain cannot even be geometrically ergodic if this essential boundedn…
Regular $g$-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories are known to be equivalent. Its equivalence for more general cases was an open problem. We present a simple example settling this issue in a negative way: there exist $g$-measures that are continuous and non-null but are not Gibbsian. Our example belongs, in fact, to a well-studied family of proc…
This article analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present a natural interpretation of these methods in terms of well known unbiasedness properties of Feynman-Kac particle measures, and a new duality with many-body Feynman-Kac models. This perspective sheds a new light on the foundations and the mathematical analysis of this class of methods. A key consequence is the equivalence between the backward and ancestral particle Markov chain Monte Carlo methods, and Gibbs sampling of a many-body …
We study the induced measure obtained from a 1-step Markov measure, supported by a topological Markov chain, after the mapping of the original alphabet onto another one. We give sufficient conditions for the induced measure to be a Gibbs measure (in the sense of Bowen) when the factor system is again a topological Markov chain. This amounts to constructing, when it does exist, the induced potential and proving its Holder continuity. This is achieved through a matrix method. We provide examples and counterexamples to illustrate our results.
We announce a new theorem bearing on high-temperature 2D Bose gases. In a certain mean-field-like regime, the grand-canonical quantum Gibbs state reduces to a nonlinear Gibbs measure constructed from a renormalized mean-field energy functional. This establishes a rigorous connection between quantum fields and (singular) classical fields in the regime of interest.
Chargaff's second parity rule (CSPR) asserts that the frequencies of short polynucleotide chains are the same as those of the complementary reversed chains. Up to now, this hypothesis has only been observed empirically and there is currently no explanation for its presence in DNA strands. Here we argue that CSPR is a probabilistic consequence of the reverse complementarity between paired strands, because the Gibbs distribution associated with the chemical energy between the bonds satisfies CSPR. We develop a statistical test to study the validity of CSPR under the Gibbsian assumption and we ap…
The Dobrushin comparison theorem is a powerful tool to bound the difference between the marginals of high-dimensional probability distributions in terms of their local specifications. Originally introduced to prove uniqueness and decay of correlations of Gibbs measures, it has been widely used in statistical mechanics as well as in the analysis of algorithms on random fields and interacting Markov chains. However, the classical comparison theorem requires validity of the Dobrushin uniqueness criterion, essentially restricting its applicability in most models to a small subset of the natural pa…
Random-scan Gibbs samplers possess a natural hierarchical structure. The structure connects Gibbs samplers targeting higher dimensional distributions to those targeting lower dimensional ones. This leads to a quasi-telescoping property of their spectral gaps. Based on this property, we derive three new bounds on the spectral gaps and convergence rates of Gibbs samplers on general domains. The three bounds relate a chain's spectral gap to, respectively, the correlation structure of the target distribution, a class of random walk chains, and a collection of influence matrices. Notably, one of ou…
Starting from the full--shift on a finite alphabet $A$, mingling some symbols of $A$, we obtain a new full shift on a smaller alphabet $B$. This amalgamation defines a factor map from $(A^{\mathbb N},T_A)$ to $(B^{\mathbb N},T_B)$, where $T_A$ and $T_B$ are the respective shift maps. According to the thermodynamic formalism, to each regular function (`potential') $ψ:A^{\mathbb N}\to{\mathbb R}$, we can associate a unique Gibbs measure $μ_ψ$. In this article, we prove that, for a large class of potentials, the pushforward measure $μ_ψ\circπ^{-1}$ is still Gibbsian for a potential $ϕ:B^{\mathbb …
We study Gibbsian models of unbounded integer-valued spins on trees which possess a symmetry under height-shift. We develop a theory relating boundary laws to gradient Gibbs measures, which applies also in cases where the corresponding Gibbs measures do not exist. Our results extend the classical theory of Zachary beyond the case of normalizable boundary laws, which implies existence of Gibbs measures, to periodic boundary laws. We provide a construction for classes of tree-automorphism invariant gradient Gibbs measures in terms of mixtures of pinned measures, whose marginals to infinite paths…
