Large deviations for the stochastic derivative Ginzburg-Landau equation with multiplicative noise

This paper first proves the existence of a unique mild solution to the stochastic derivative Ginzburg-Landau equation. The fixed point theorem for the corresponding truncated equation is used as the main tool. Since we restrict our study to the one-dimensional case, it is not necessary to introduce another Banach space and thus the estimates of the stochastic convolutions in the Banach space are avoided. Secondly, we also consider large deviations for the stochastic derivative Ginzburg-Landau equation perturbed by a small noise. Since the underlying space considered is Polish, using the weak convergence approach, we establish a large deviations principle by proving a Laplace principle.     

Large deviations for ℤ d -Actions

We establish large deviations bounds for translation invariant Gibbs measures of multidimensional subshifts of finite type. This generalizes [FO] and partially [C, O, and B], where only full shifts were considered. Our framework includes, in particular, the hard-core lattice gas models which are outside of the scope of [FO, C, O, and B].Partially supported by US-Israel BSF. Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Large deviations for stationary Gaussian processes

In their previous work on large deviations the authors always assumed the base process to be Markovian whereas here they consider the base process to be stationary Gaussian. Similar large deviation results are obtained under natural hypotheses on the spectral density function of the base process. A rather explicit formula for the entropy involved is also obtained.The research in this paper was supported by the National Science Foundation, Grant No. MCS-80-02568     

Large deviations for noninteracting infinite-particle systems

A large deviation property is established for noninteracting infinite particle systems. Previous large deviation results obtained by the authors involved a singleI-function because the cases treated always involved a unique invariant measure for the process. In the context of this paper there is an infinite family of invariant measures and a corresponding infinite family ofI-functions governing the large deviations.     

Large deviations for Ising spin glasses with constrained disorder

We consider ad-dimensional Ising spin glass and construct lower bounds for the mean free energy density which, in general, improve the classical lower bounds given by the annealed free energy density. The bounds are achieved by introducing generalized finite-volume free energy densities. The large-deviations aspects of the problem are displayed and examples discussed.     

Large deviations in spin-glass ground-state energies

Many crystalline networks can be viewed as decorations of triply periodic minimal surfaces. Such surfaces are covered by the hyperbolic plane in the same way that the Euclidean plane covers a cylinder. Thus, a symmetric hyperbolic network can be wrapped onto an appropriate minimal surface to obtain a 3d periodic net. This requires symmetries of the hyperbolic net to match the symmetries of the minimal surface. We describe a systematic algorithm to find all the hyperbolic symmetries that are commensurate with a given minimal surface, and the generation of simple 3d nets from these symmetry groups.PACS: 61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling - 89.75.Hc Networks and genealogical trees - 02.20.-a Group theory - 02.40.-k Geometry, differential geometry, and topologyThis revised version was published online in July 2004. It contains colour changes to Figure 6.     

Revealing the building blocks of spatiotemporal chaos: deviations from extensivity

We have performed high-precision computational studies of the fractal dimension as a function of system length for spatiotemporal chaotic states of the one-dimensional complex Ginzburg-Landau equation. Our data show deviations from extensivity on a length scale consistent with the chaotic length scale, indicating that this spatiotemporal chaotic system is composed of weakly interacting building blocks, each containing about 2 degrees of freedom. Our results also suggest an explanation of some of the "windows of periodicity" found in spatiotemporal systems of moderate size.     

Large deviations estimates for the multiscale analysis of heart rate variability

In the realm of multiscale signal analysis, multifractal analysis provides a natural and rich framework to measure the roughness of a time series. As such, it has drawn special attention of both mathematicians and practitioners, and led them to characterize relevant physiological factors impacting the heart rate variability. Notwithstanding these considerable progresses, multifractal analysis almost exclusively developed around the concept of Legendre singularity spectrum, for which efficient and elaborate estimators exist, but which are structurally blind to subtle features like non-concavity or, to a certain extent, non scaling of the distributions. Large deviations theory allows bypassing these limitations but it is only very recently that performing estimators were proposed to reliably compute the corresponding large deviations singularity spectrum. In this article, we illustrate the relevance of this approach, on both theoretical objects and on human heart rate signals from the Physionet public database. As conjectured, we verify that large deviations principles reveal significant information that otherwise remains hidden with classical approaches, and which can be reminiscent of some physiological characteristics. In particular we quantify the presence/absence of scale invariance of RR signals.     

Large deviations of extreme eigenvalues of random matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N x N) random matrix are positive (negative) decreases for large N as approximately exp[-betatheta(0)N2] where the parameter beta characterizes the ensemble and the exponent theta(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at zeta.     

Large deviations and the distribution of pre-images of rational maps

In this article we prove a large deviation result for the pre-images of a point in the Julia set of a rational mapping of the Riemann sphere. As a corollary, we deduce a convergence result for certain weighted averages of orbital measures, generalizing a result of Lyubich.The first author was supported by a Royal Society University Fellowship during part of this research.     

Large deviations in the free energy of mean-field spin glasses

We compute analytically the probability distribution of large deviations in the spin-glass free energy for the Sherrington-Kirkpatrick mean-field model; i.e., we compute the exponentially small probability of finding a system with intensive free energy smaller than the most likely one. This result is obtained by computing the average value of the partition function to the power n as a function of n. At zero temperature this absolute prediction displays a remarkable quantitative agreement with the numerical data.     

Large deviations,averaging and periodic orbits of dynamical systems

The paper deals with large deviation bounds for the proportion of periodic orbits with irregular behavior for expansive dynamical systems with specification, in particular, we obtain estimates for large deviations from the equidistribution for closed geodesics on negatively curved manifolds. We derive also large deviation bounds in the averaging principle when the fast motion is the shift along periodic orbits.Partially supported by US-Israel BSFPartially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation, (Germany)     

Large deviations for the 2D ising model: A lower bound without cluster expansions

We show that a lower large-deviation bound for the block-spin magnetization in the 2D Ising model can be pushed all the way forward toward its correct Wulff value for all >_c.     

Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians

We consider the integrated density of states (IDS) ρ_∞(λ) of random Hamiltonian H_ω=?Δ+V_ω, V_ω being a random field on ? ^d which satisfies a mixing condition. We prove that the probability of large fluctuations of the finite volume IDS |Λ|^?1ρ(λ, H_Λ(ω)), Λ ? ? ^d , around the thermodynamic limit ρ_∞(λ) is bounded from above by exp {?k|Λ|},k>0. In this case ρ_∞(λ) can be recovered from a variational principle. Furthermore we show the existence of a Lifshitztype of singularity of ρ_∞(λ) as λ → 0⁺ in the case where V_ω is non-negative. More precisely we prove the following bound: ρ_∞(λ)≦exp(?kλ^?d/2) as λ → 0⁺ k>0. This last result is then discussed in some examples.     

Large deviations from classical paths. Hamiltonian flows as classical limits of quantum flows

We prove that in the limit0, the probability for the paths of the stochastic jump process associated to the quantum time evolution to be in a tublet around the classical trajectory is of order 1–exp{–A/}. We give some applications of this result to the study of the classical limit of Wigner functions.     

An equation for continuous chaos

A prototype equation to the Lorenz model of turbulence contains just one (second-order) nonlinearity in one variable. The flow in state space allows for a “folded” Poincaré map (horseshoe map). Many more natural and artificial systems are governed by this type of equation.     

Chapman-Kolmogorov equation for discrete chaos

We derive an equation of the Chapman-Kolmogorov type for multi-dimensional discrete mappings under the impact of additive and multiplicative noise with arbitrary distribution.