Large Deviations and Scaling Limit

We explore the behavior under scaling limits of large systems using methods from the theory large deviations. This is carried out through the examination of a few examples.     

Adaptive Sampling of Large Deviations

We introduce and test an algorithm that adaptively estimates large deviation functions characterizing the fluctuations of additive functionals of Markov processes in the long-time limit. These functions play an important role for predicting the probability and pathways of rare events in stochastic processes, as well as for understanding the physics of nonequilibrium systems driven in steady states by external forces and reservoirs. The algorithm uses methods from risk-sensitive and feedback control to estimate from a single trajectory a new process, called the driven process, known to be efficient for importance sampling. Its advantages compared to other simulation techniques, such as splitting or cloning, are discussed and illustrated with simple equilibrium and nonequilibrium diffusion models.     

Large Deviations for Random Trees

We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.     

The Yang-Yang thermodynamic formalism and Large Deviations

The partition function for a one-dimensional system of Bosons with repulsive delta-function interaction is investigated. We prove that if the Bethe Ansatz eigenfunctions form a complete set then the grand canonical pressure is given by the Yang-Yang formula. The proof uses a probabilistic formalism to express the partition function as an expectation with respect to a probability measure on a Banach space of measures; the asymptotic behaviour of the expectation in the thermodynamic limit is determined by the Large Deviation Principle. This method is applicable in situations in which the Hamiltonian can be diagonalised using the Bethe Ansatz.     

Large Deviations,Fluctuations and Shrinking Intervals

This paper concerns the statistical properties of hyperbolic diffeomorphisms. We obtain a large deviation result with respect to slowly shrinking intervals for a large class of Hölder continuous functions. In case of time reversal symmetry, we obtain a corresponding version of the Fluctuation Theorem.     

Large Deviations for Quantum Spin Systems

We consider high temperature KMS states for quantum spin systems on a lattice. We prove a large deviation principle for the distribution of empirical averages , where the X _i 's are copies of a self-adjoint element X (level one large deviations). From the analyticity of the generating function, we obtain the central limit theorem. We generalize to a level two large deviation principle for the distribution of      

Large Deviations for Non-Uniformly Expanding Maps

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. 2000 Mathematics Subject Classification: 37D25, 37A50, 37B40, 37C40     

Large Deviations for Probabilistic Cellular Automata

We consider a generalized model of a probabilistic cellular automata described by a Markov chain on an infinite dimensional space and derive certain large deviations bounds for corresponding occupational measures.     

Large Deviations in Rarefied Quantum Gases

The probability of observing a large deviation (LD) in the number of particles in a region in a dilute quantum gas contained in a much larger region V is shown to decay as exp[–||F], where || is the volume of and F is the change in the appropriate free energy density, the same as in classical systems. However, in contrast with the classical case, where this formula holds at all temperatures and chemical potentials our proof is restricted to rarefied gases, both for the typical and observed density, at least for Bose or Fermi systems. The case of Boltzmann statistics with a bounded repulsive potential can be treated at all temperatures and densities. Fermions on a lattice in any dimension, or in the continuum in one dimension, can be treated at all densities and temperatures if the interaction is small enough (depending on density and temperature), provided one assumes periodic boundary conditions.     

Large Deviations in Quantum Spin Chains

We show the full large deviation principle for KMS-states and C*-finitely correlated states on a quantum spin chain. We cover general local observables. Our main tool is Ruelle’s transfer operator method.     

Large Deviations and Ergodicity for Spin Particle Systems

In this paper we investigate the large deviation principle (LDP) for spin particle systems with possibly vanishing flip rates. The situation turns out to be much more complicated if the flip rates are allowed to be zero than the one considered by Dai, where the systems are assumed to have strictly positive flip rates. The upper and lower large-deviation bounds are studied, respectively. The two governing rate functions are compared and a variational principle is given. We then apply the results to obtain some new large-deviation estimates for the occupation times of attractive systems. In particular, we prove a strong form of exponential convergence for ergodic systems.     

Large Deviations for Probabilistic Cellular Automata II

We obtain an upper large deviations bound which shows that for some models of probabilistic cellular automata (which are far away from the product case) the lower large deviation bound derived in Eizenberg and Kifer J. Stat. Phys. 108: 1255–1280 (2002) is sharp, and so the corresponding large deviations phenomena cannot be described via the traditional Donsker–Varadhan form of the action functional. For models which are close to the product case we derive approximate large deviations bounds using the Donsker–Varadhan functional for the product case.     

Large Deviations for Gaussian Diffusions with Delay

Dynamical systems driven by nonlinear delay SDEs with small noise can exhibit important rare events on long timescales. When there is no delay, classical large deviations theory quantifies rare events such as escapes from metastable fixed points. Near such fixed points, one can approximate nonlinear delay SDEs by linear delay SDEs. Here, we develop a fully explicit large deviations framework for (necessarily Gaussian) processes \(X_t\) driven by linear delay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional controlling rare events for \(X_t\) and of the most likely paths transiting from \(X_0 = p\) to \(X_T=q\). Via linear noise local approximations, we can then compute most likely routes of escape from metastable states for nonlinear delay SDEs. We apply our methodology to the detailed dynamics of a genetic regulatory circuit, namely the co-repressive toggle switch, which may be described by a nonlinear chemical Langevin SDE with delay.     

Correction to: Adaptive Sampling of Large Deviations

Journal of Statistical Physics - The original version of this article unfortunately contained an error. The authors would like to correct the error with this erratum.     

Large Deviations and a Fluctuation Symmetry for Chaotic Homeomorphisms

 We consider expansive homeomorphisms with the specification property. We give a new simple proof of a large deviation principle for Gibbs measures corresponding to a regular potential and we establish a general symmetry of the rate function for the large deviations of the antisymmetric part, under time-reversal, of the potential. This generalizes the Gallavotti-Cohen fluctuation theorem to a larger class of chaotic systems. Received: 6 March 2002 / Accepted: 16 September 2002 Published online: 8 January 2003     

Non-uniform Specification and Large Deviations for Weak Gibbs Measures

We establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak Gibbs measures. Under some mild assumptions, we obtain upper and lower bounds for the measure of deviation sets of some non-uniformly expanding maps, including quadratic maps and robust multidimensional non-uniformly expanding local diffeomorphisms. For that purpose, a measure theoretical weak form of specification is introduced and proved to hold for the robust classes of multidimensional non-uniformly expanding local diffeomorphisms and Viana maps.     

On the Estimation of the Large Deviations Spectrum

We propose an estimation algorithm for large deviations spectra of measures and functions. The algorithm converges for natural examples of multifractals.     

EPR States and Bell Correlated States in Algebraic Quantum Field Theory

A mathematical rigorous definition of EPR states has been introduced by Arens and Varadarajan for finite dimensional systems, and extended by Werner to general systems. In the present paper we follow a definition of EPR states due to Werner. Then we show that an EPR state for incommensurable pairs is Bell correlated, and that the set of EPR states for incommensurable pairs is norm dense between two strictly space-like separated regions in algebraic quantum field theory.