Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

Summary For Gibbsian systems of particles inR ^d , we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.     

Multidimensional random subshifts of finite type and their large deviations

Summary I introduce random multidimensional subshifts of finite type which generalize models of spin-glasses and establish the “almost sure” large deviations bounds for Gibbs measures there. The paper is sequel to [EKW] where the corresponding results were obtained for deterministic multidimensional subshifts of finite type. Partially supported by US-Israel BSF     

Large deviations for random walks in a mixing random environment and other (non‐Markov) random walks

We extend a recent work by S. R. S. Varadhan [8] on large deviations for random walks in a product random environment to include more general random walks on the lattice. In particular, some reinforced random walks and several classes of random walks in Gibbs fields are included. © 2004 Wiley Periodicals, Inc.     

Methode de laplace: etude variationnelle des fluctuations de diffusions de type

This article concerns the asymptotic behaviour of diffusions consisting of various systems of mean-field type interacting particles. We specifically study the following properties: propagation of chaos; critical and non critical fluctuations, using the Laplace method which has been developed by Bolthausen in the general Banach valued random variables context; and finally the large deviations of laws of trajectorial empirical measures.

We also study Gibbs variational formula associated with this problem and show that it reduces to a finite dimensional problem     

On evaluating the rate function of large deviations for jump processes

This is a sequel to our joint paper^[4] in which upper bound estimates for large deviations for Markov chains are studied. The purpose of this paper is to characterize the rate function of large deviations for jump processes. In particular, an explicit expression of the rate function is given in the case of the process being symmetrizable.     

Refinements of the Gibbs conditioning principle

Summary Refinements of Sanov's large deviations theorem lead via Csiszár's information theoretic identity to refinements of the Gibbs conditioning principle which are valid for blocks whose length increase with the length of the conditioning sequence. Sharp bounds on the growth of the block length with the length of the conditioning sequence are derived.Partially supported by NSF DMS92-09712 grant and by a US-Israel BSF grantPartially supported by a US-Israel BSF grant and by the fund for promotion of research at the Technion     

Quenched asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials

 This article describes the almost sure infinite volume asymptotics of the ground state energy of random Schr?dinger operators with scaled Gibbsian potentials. The random potential is obtained by distributing soft obstacles according to an infinite volume grand canonical tempered Gibbs measure with a superstable pair interaction. There is no restriction on the strength of the pair interaction: it may be taken, e.g., at a critical point. The potential is scaled with the box size in a critical way, i.e. the scale is determined by the typical size of large deviations in the Gibbsian cloud. The almost sure infinite volume asymptotics of the ground state energy is described in terms of two equivalent deterministic variational principles involving only thermodynamic quantities. The qualitative behaviour of the ground state energy asymptotics is analysed: Depending on the dimension and on the H?lder exponents of the free energy density, it is identified which cases lead to a phase transition of the asymptotic behaviour of the ground state energy. Received: 24 June 2002 / Revised version: 17 February 2003 Published online: 12 May 2003 Mathematics Subject Classification (2000): Primary 82B44; Secondary 60K35 Key words or phrases: Gibbs measure – H?lder exponents – Random Schr?dinger operator – Ground state – Large deviations     

Pseudo-free energies and large deviations for non gibbsian FKG measures

Summary A large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on {0, 1{ ^{Z d} is proven. In this way a pseudo-free energy and pressure is defined. For ergodic systems the large deviations property holds with the usual scaling. The case of non ergodic systems is also discussed. A similar result holds for occupation times. The perturbation by an external field is treated.Work partially supported by NSF-DMR81-14726 (USA) and CNPq (Brazil)Also Department of Physics     

The Gibbs Phenomenon for n-Dimensional Kernels

We give conditions which ensure the existence in Rⁿ of the Gibbs phenomenon for a large class of kernels non necessarily rotation or translation invariant.     

Rate function of large deviation for a class of nonhomogeneous markov chains on supercritical percolation network

We model an epidemic with a class of nonhomogeneous Markov chains on the supercritical percolation network on ℤ ^d . The large deviations law for the Markov chain is given. Explicit expression of the rate function for large deviation is obtained.     

A Large Deviation Principle for the free energy of random Gibbs measures with application to the REM

A Large Deviation Principle (LDP) for the free energy of random Gibbs measures is proved in the form of a general LDP for random log-Laplace integrals. The principle is then applied to an extended version of the Random Energy Model (REM). The rate of exponential decay for the classical REM is stronger than the known concentration exponent, and probabilities of negative deviations are super-exponentially small.     

Convergence rate of gibbs sampler and its application

Based on the convergence rate defined by the Pearson-χ~2 distance,this pa- per discusses properties of different Gibbs sampling schemes.Under a set of regularity conditions,it is proved in this paper that the rate of convergence on systematic scan Gibbs samplers is the norm of a forward operator.We also discuss that the collapsed Gibbs sam- pler has a faster convergence rate than the systematic scan Gibbs sampler as proposed by Liu et al.Based on the definition of convergence rate of the Pearson-χ~2 distance, this paper proved this result quantitatively.According to Theorem 2,we also proved that the convergence rate defined with the spectral radius of matrix by Robert and Shau is equivalent to the corresponding radius of the forward operator.     

Construction of Gibbs measures for 1-dimensional continuum fields

We study 1-dimensional continuum fields of Ginzburg-Landau type under the presence of an external and a long-range pair interaction potentials. The corresponding Gibbs states are formulated as Gibbs measures relative to Brownian motion [17]. In this context we prove the existence of Gibbs measures for a wide class of potentials including a singular external potential as hard-wall ones, as well as a non-convex interaction. Our basic methods are: (i) to derive moment estimates via integration by parts; and (ii) in its finite-volume construction, to represent the hard-wall Gibbs measure on C(ℝ;ℝ⁺) in terms of a certain rotationally invariant Gibbs measure on C(ℝ;ℝ³).     

Stochastic Dynamics of Compact Spins: Ergodicity and Irreducibility

Stochastic dynamics associated with Gibbs measures on M ^{Z d} , where M is a compact Riemannian manifold and Z ^d is an integer lattice, is considered. Equivalence of its L ²-ergodicity and the extremality of the corresponding Gibbs measure is proved.     

Large deviations and stochastic homogenization

Summary A general theorem is stated providing large deviations estimates for a family of measures on a topological vector space. Applications are given in the second part, where large deviations problems arising in stochastic homogenization are discussed. Another application is given in similar problems connected with Donsker's invariance principle.     

次线性期望下独立随机变量列的大偏差和中偏差

本文研究在次线性期望下的独立随机变量列的大偏差和中偏差原理. 利用次可加方法, 我们得 到次线性期望下的大偏差原理. 与次线性期望下的中心极限定理相应的中偏差原理也被建立.     

Large deviations principles for stochastic scalar conservation laws

Large deviations principles for a family of scalar 1 + 1 dimensional conservative stochastic PDEs (viscous conservation laws) are investigated, in the limit of jointly vanishing noise and viscosity. A first large deviations principle is obtained in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. A second order large deviations principle is therefore investigated, however, this can be only partially proved. The second order rate functional provides a generalization for non-convex fluxes of the functional introduced by Jensen and Varadhan in a stochastic particles system setting.     

Large deviations for empirical measures with degenerate limiting distribution

Summary This paper studies the large deviations of the empirical measure associated withn independent random variables with a degenerate limiting distribution asn. A large deviations principle — quite unlike the classical Sanov type results — is established for such empirical measures in a general Polish space setting. This result is applied to the large deviations for the empirical process of a system of interacting particles, in which the diffusion coefficient vanishes as the number of particles tends to infinity. A second way in which the present example differs from previous work on similar weakly interacting systems is that there is a singularity in the mean-field type interaction.     

Shannon–McMillan theorems for discrete random fields along curves and lower bounds for surface-order large deviations

The notion of a surface-order specific entropy h _c (P) of a two-dimensional discrete random field P along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c. Existence is shown by proving a corresponding Shannon–McMillan theorem. We obtain a representation of h _c (P) as a mixture of specific entropies along the tangent lines of c. As an application, the specific entropy along curves is used to refine Föllmer and Ort’s lower bound for the large deviations of the empirical field of an attractive Gibbs measure from its ergodic behaviour in the phase-transition regime.     

Weakly coupled Gibbs measures

Summary We formulate an abstract functional-analytic framework for the study of Gibbs measures on infinite product spaces. Working in this frame-work, we present a detailed analysis of the weak-coupling regime. Specifically, we derive general theorems on existence of the Gibbs measure, analyticity in its component Gibbs factors, and exponential decay of correlations and truncated expectations in the spread of distant families of random variables. In translation-invariant situations we obtain a central limit theorem. Our main tool is a series expansion in truncated expectations, which we analyze with L ^p methods.Original title: Analyticity and Decay of Correlations in Weakly Coupled Lattice Models.Supported by N.S.F. Grant PHY76-17191Dedicated to Professor Leopold Schmetterer