Robust Exponential Decay of Correlations for Singular-Flows

We construct open sets of C ^k (k ≥ 2) vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure. In particular we prove that the geometric Lorenz attractor has exponential decay of correlations with respect to the unique physical measure.     

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuous-time dynamics with singularities on a manifold. Our proof combines the second author’s version (Liverani in Ann Math 159:1275–1312, 2004) of Dolgopyat’s estimates for contact flows and the first author’s work with Gouëzel (J Mod Dyn 4:91–137, 2010) on piecewise hyperbolic discrete-time dynamics.     

Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata

We investigate the low-noise regime of a large class of probabilistic cellular automata, including the North-East-Center model of Toom. They are defined as stochastic perturbations of cellular automata belonging to the category of monotonic binary tessellations and possessing a property of erosion. We prove, for a set of initial conditions, exponential convergence of the induced processes toward an extremal invariant measure with a highly predominant spin value. We also show that this invariant measure presents exponential decay of correlations in space and in time and is therefore strongly mixing.     

Decay of Correlations and Dispersing Billiards

We give a rigorous proof of exponential decay of correlations for all major classes of planar dispersing billiards: periodic Lorentz gases with and without horizon and dispersing billiard tables with corner points     

Decay of Correlations in Random-Cluster Models

We prove exponential decay for the tail of the radius R of the cluster at the origin, for subcritical random-cluster models, under an assumption slightly weaker than that (here, d is the number of dimensions). Specifically, if throughout the subcritical phase, then for some α > 0. This implies the exponential decay of the two-point correlation function of subcritical Potts models, subject to a hypothesis of (at least) polynomial decay of this function. Similar results are known already for percolation and Ising models, and for Potts models when the number q of available states is sufficiently large; indeed the hypothesis of polynomial decay has been proved rigorously for these cases. In two dimensions, the hypothesis that is weaker than requiring that the susceptibility be finite, i.e., that the two-point function be summable. The principal new technique is a form of Russo's formula for random-cluster models reported by Bezuidenhout, Grimmett, and Kesten. For the current application, this leads to an analysis of a first-passage problem for random-cluster models, and a proof that the associated time constant is strictly positive if and only if the tail of R decays exponentially. Received: 25 September 1996 / Accepted: 21 February 1997     

Quantum Stabilization and Decay of Correlations in Anharmonic Crystals

Letters in Mathematical Physics -     

Liapunov Multipliers and Decay of Correlations in Dynamical Systems

The essential decorrelation rate of a hyperbolic dynamical system is the decay rate of time-correlations one expects to see stably for typical observables once resonances are projected out. We define and illustrate these notions and study the conjecture that for observables in $\mathcal{C}^1$ , the essential decorrelation rate is never faster than what is dictated by the smallest unstable Liapunov multiplier.     

Witten Laplacian Method for The Decay of Correlations

The aim of this paper is to apply direct methods to the study of integrals that appear naturally in Statistical Mechanics and Euclidean Field Theory. We provide weighted estimates leading to the exponential decay of the two-point correlation functions for certain classical convex unbounded models. The methods involve the study of the solutions of the Witten Laplacian equations associated with the Hamiltonian of the system.     

Exponential Decay of Dispersion-Managed Solitons for General Dispersion Profiles

We show that any weak solution of the dispersion management equation describing dispersion-managed solitons together with its Fourier transform decay exponentially. This strong regularity result extends a recent result of Erdo?an, Hundertmark, and Lee in two directions, to arbitrary non-negative average dispersion and, more importantly, to rather general dispersion profiles, which cover most, if not all, physically relevant cases.     

Computer-Assisted Bounds for the Rate of Decay of Correlations

The rate of decay of correlations quantitatively describes the rate at which a chaotic system “mixes” the state space. We present a new rigorous method to estimate a bound for this rate of mixing. The technique may be implemented on a computer and is applicable to both multidimensional expanding and hyperbolic systems. The bounds produced are significantly less conservative than current rigorous bounds. In some situations it is possible to approximate resonant eigenfunctions and to strengthen our bound to an estimate of the decay rate. Order of convergence results are stated. Received: 8 January 1997 / Accepted: 20 March 1997     

Exponential Decay Towards Equilibrium for the Inhomogeneous Aizenman-Bak Model

The Aizenman-Bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a non-degenerate size-dependent coefficient. Both the break-up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels. We demonstrate that the entropy-entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension, but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non-degenerate diffusion in dimension one. We finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial. Up to our knowledge, this is the first result of explicit equilibration rates for spatially inhomogeneous coagulation-fragmentation models.     

Dispersing Billiards with Cusps: Slow Decay of Correlations

Dispersing billiards introduced by Sinai are uniformly hyperbolic and have strong statistical properties (exponential decay of correlations and various limit theorems). However, if the billiard table has cusps (corner points with zero interior angles), then its hyperbolicity is nonuniform and statistical properties deteriorate. Until now only heuristic and experimental results existed predicting the decay of correlations as . We present a first rigorous analysis of correlations for dispersing billiards with cusps.     

Spectral Spacing Correlations for Chaotic and Disordered Systems

New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point correlation function. Their behavior results from two different contributions. One corresponds to (universal) random matrix eigenvalue fluctuations, the other to diffusive or chaotic characteristics of the corresponding classical motion. A closed formula expressing spacing autocovariances in terms of classical dynamical zeta functions, including the Perron–Frobenius operator, is derived. It leads to a simple interpretation in terms of classical resonances. The theory is applied to zeros of the Riemann zeta function. A striking correspondence between the associated classical dynamical zeta functions and the Riemann zeta itself is found. This induces a resurgence phenomenon where the lowest Riemann zeros appear replicated an infinite number of times as resonances and sub-resonances in the spacing autocovariances. The theoretical results are confirmed by existing data. The present work further extends the already well known semiclassical interpretation of properties of Riemann zeros.     

Spectral Gap for Multi-species Exclusion Processes

We consider a multi-species generalization of the symmetric simple exclusion process in homogeneous and non-homogeneous hypercubes of Z ^d . In this model, the hyperplanes of configurations with given numbers of particles of each species are not necessarily irreducible. We give a sufficient condition of the dynamics to make them irreducible. In addition, assuming the irreducibility of them, we show some estimates of the spectral gap (the absolute value of the second largest eigenvalue of the generator), which plays an important role in the study of hydrodynamic limit.     

Spectral Gap of Segments of Periodic Waveguides

The lowest spectral gap of segments of a periodic waveguide in is proportional to the square of the inverse length. Dedicated to Pavel Exner on the occasion of his 60th birthday.     

An Invariance Principle for Maps with Polynomial Decay of Correlations

We give a general method of deriving statistical limit theorems, such as the central limit theorem and its functional version, in the setting of ergodic measure preserving transformations. This method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations.     

The Spectral Gap of the Ferromagnetic XXZ-Chain

We prove that the spectral gap of the spin- ferromagnetic XXZ-chain with HamiltonianH=–_x S^{(1)}_xS^{(1)}_{x+1}+S^{(2)}_xS^{(2)}_{x+1}+\Delta S^{(3)}_xS^{(3)}_{x+1}, is given by -1 for all 1. This is the gap in the spectrum of the infinite chainin any of its ground states, the translation invariant ones as well asthe kink ground states, which contain an interface between an up and a down region.In particular, this shows that the lowest magnon energy is not affected by the presence of a domain wall. This surprising fact is a consequence of the SU _q (2)quantum group symmetry of the model.