Related papers: Introduction to (generalized) Gibbs measures
We consider some of the main notions of Gibbs measures on subshifts introduced by different communities, such as dynamical systems, probability, operator algebras, and mathematical physics. For potentials with $d$-summable variation, we…
Lecture notes for a minicourse to given in the XVII Brazilian School of Geometry, UFAM (Amazonas), Brazil, July 2012.
The notion of Gibbs Measure is used by many researchers of the communities of Mathematical Physics, Probability, Thermodynamic Formalism, Symbolic Dynamics, and others. A natural question is when these several different notions of Gibbs…
We present a novel approach to establishing the variational principle for Gibbs and generalized (weak and almost) Gibbs states. Limitations of a thermodynamical formalism for generalized Gibbs states will be discussed. A new class of…
Mathematical models in equilibrium statistical mechanics describe physical systems with many particles interacting with an external force and with one another. Gibbs measure is a fundamental concept in this theory. In existing literature…
We prove a priori estimates and, as sequel, existence of Euclidean Gibbs states for quantum lattice systems. For this purpose we develop a new analytical approach, the main tools of which are: first, a characterization of the Gibbs states…
This paper studies the Gibbs measure of an interacting particle system with a general interaction kernel at various temperature regimes. We are particularly interested in fine features of the convergence to the mean-field density as the…
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years,…
Models of quantum and classical particles on the d-dimensional cubic lattice with pair interparticle interactions are considered. The classical model is obtained from the corresponding quantum one when the reduced physical mass of the…
Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allowed to…
The questions of justification of the Gibbs canonical distribution for systems with elastic impacts are discussed. A special attention is paid to the description of probability measures with densities depending on the system energy.
We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the…
We consider a gas whose each particle is characterised by a pair $(x,v_x)$ with the position $x\in \mathbb R^d$ and the velocity $v_x\in \mathbb R^d_0= \mathbb R^d\setminus \{0\}$. We define Gibbs measures on the cone of vector-valued…
Gibbs measure is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random…
We study the thermodynamic formalism for generalized Gibbs measures, such as renormalization group transformations of Gibbs measures or joint measures of disordered spin systems. We first show existence of the relative entropy density and…
The present overview and gentle introduction to Grassmannian calculus and some of its applications to probability collects the notes of a mini-course given by the authors at the Brazilian School of Probability, August 5-9, 2024, in…
In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of…
We study a class of Gibbs measures of classical particle spin systems with spin space $S=\mathbb{R}^{m}$ and unbounded pair interaction, living on a metric graph given by a typical realization $\gamma $ of a random point process in…
A lattice system of interacting temperature loops, which is used in the Euclidean approach to describe equilibrium thermodynamic properties of an infinite system of interacting quantum particles performing anharmonic oscillations (quantum…
We study the relative entropy density for generalized Gibbs measures. We first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of ``almost Gibbsian measures'' (almost sure continuity…