Searcharxiv⌕ Search

arXiv subjects

J. -R. Chazottes

Publications and source records attributed to J. -R. Chazottes.

At least 37 records · Page 2Linked to original sources
arXiv2017

On concentration inequalities and their applications for Gibbs measures in lattice systems

We consider Gibbs measures on the configuration space $S^{\mathbb{Z}^d}$, where mostly $d\geq 2$ and $S$ is a finite set. We start by a short review on concentration inequalities for Gibbs measures. In the Dobrushin uniqueness regime, we have a Gaussian concentration bound, whereas in the Ising model (and related models) at sufficiently low temperature, we control all moments and have a stretched-exponential concentration bound. We then give several applications of these inequalities whereby we obtain various new results. Amongst these applications, we get bounds on the speed of convergence of the empirical measure in the sense of Kantorovich distance, fluctuation bounds in the Shannon-McMillan-Breiman theorem, fluctuation bounds for the first occurrence of a pattern, as well as almost-sure central limit theorems.

math.PR↗
arXiv2014

Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes

We study a general class of birth-and-death processes with state space $\mathbb{N}$ that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is measured in terms of a `carrying capacity' $K$. When $K$ is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the behavior of the process and the mean time to extinction in the quasi-stationary distribution as a function of $K$, for large $K$. We also give a quantitative description of this quasi-stationary distribution. It turns out to be close to a Gaussian distribution centered about the deterministic long-time equilibrium, when $K$ is large. Our analysis relies on precise estimates of the maximal eigenvalue, of the corresponding eigenvector and of the spectral gap of a self-adjoint operator associated with the semigroup of the process.

math.PR↗
arXiv2014

Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems

We introduce the multiplicative Ising model and prove basic properties of its thermodynamic formalism such as existence of pressure and entropies. We generalize to one-dimensional "layer-unique" Gibbs measures for which the same results can be obtained. For more general models associated to a $d$-dimensional multiplicative invariant potential, we prove a large deviation theorem in the uniqueness regime for averages of multiplicative shifts of general local functions. This thermodynamic formalism is motivated by the statistical properties of multiple ergodic averages.

math.PR↗
arXiv2013

Tilings of the plane and Thurston semi-norm

We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a finite-dimensional simplex. This function is a generalisation, in the framework of branched surfaces, of the Thurston semi-norm originally defined for compact $3$-manifolds.

math.MG↗
arXiv2012

Nonconventional averages along arithmetic progressions and lattice spin systems

We study the so-called nonconventional averages in the context of lattice spin systems, or equivalently random colourings of the integers. For i.i.d. colourings, we prove a large deviation principle for the number of monochromatic arithmetic progressions of size two in the box $[1,N]\cap \N$, as $N\to\infty$, with an explicit rate function related to the one-dimensional Ising model. For more general colourings, we prove some bounds for the number of monochromatic arithmetic progressions of arbitrary size, as well as for the maximal progression inside the box $[1,N]\cap \N$. Finally, we relate nonconventional sums along arithmetic progressions of size greater than two to statistical mechanics models in dimension larger than one.

math.PR↗
arXiv2012

Fluctuations of observables in dynamical systems: from limit theorems to concentration inequalities

We start by reviewing recent probabilistic results on ergodic sums in a large class of (non-uniformly) hyperbolic dynamical systems. Namely, we describe the central limit theorem, the almost-sure convergence to the gaussian and other stable laws, and large deviations. Next, we describe a new branch in the study of probabilistic properties of dynamical systems, namely concentration inequalities. They allow to describe the fluctuations of very general observables and to get bounds rather than limit laws. We end up with two sections: one gathering various open problems, notably on random dynamical systems, coupled map lattices and the so-called nonconventional ergodic averages; and another one giving pointers to the literature about moderate deviations, almost-sure invariance principle, etc.

math.DS↗
arXiv2011

Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

Let $A$ be a finite set and $ϕ:A^Z\to R$ be a locally constant potential. For each $β>0$ ("inverse temperature"), there is a unique Gibbs measure $μ_{βϕ}$. We prove that, as $β\to+\infty$, the family $(μ_{βϕ})_{β>0}$ converges (in weak-$^*$ topology) to a measure we characterize. It is concentrated on a certain subshift of finite type which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron-Frobenius Theorem for matrices á la Birkhoff. The crucial idea we bring is a "renormalization" procedure which explains convergence and provides a recursive algorithm to compute the weights of the ergodic decomposition of the limit.

math.DS↗
arXiv2011

Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems

We study the number of visits to balls B_r(x), up to time t/mu(B_r(x)), for a class of non-uniformly hyperbolic dynamical systems, where mu is the SRB measure. Outside a set of `bad' centers x, we prove that this number is approximately Poissonnian with a controlled error term. In particular, when r-->0, we get convergence to the Poisson law for a set of centers of mu-measure one. Our theorem applies for instance to the Hénon attractor and, more generally, to systems modelled by a Young tower whose return-time function has a exponential tail and with one-dimensional unstable manifolds. Along the way, we prove an abstract Poisson approximation result of independent interest.

math.DS↗
arXiv2011

On the finite-dimensional marginals of shift-invariant measures

Let $Σ$ be a finite alphabet, $Ω=Σ^{\mathbb{Z}^{d}}$ equipped with the shift action, and $\mathcal{I}$ the simplex of shift-invariant measures on $Ω$. We study the relation between the restriction $\mathcal{I}_n$ of $\mathcal{I}$ to the finite cubes $\{-n,...,n\}^d\subset\mathbb{Z}^d$, and the polytope of "locally invariant" measures $\mathcal{I}_n^{loc}$. We are especially interested in the geometry of the convex set $\mathcal{I}_n$ which turns out to be strikingly different when $d=1$ and when $d\geq 2$. A major role is played by shifts of finite type which are naturally identified with faces of $\mathcal{I}_n$, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of $\mathcal{I}_n$, although in dimension $d\geq 2$ there are also extreme points which arise in other ways. We show that $\mathcal{I}_n=\mathcal{I}_n^{loc}$ when $d=1$, but in higher dimension they differ for $n$ large enough. We also show that while in dimension one $\mathcal{I}_n$ are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of $\mathcal{I}_n$ for all large enough $n$.

math.DS↗
arXiv2011

Poincaré inequality for Markov random fields via disagreement percolation

We consider Markov random fields of discrete spins on the lattice $\Zd$. We use a technique of coupling of conditional distributions. If under the coupling the disagreement cluster is "sufficiently" subcritical, then we prove the Poincaré inequality. In the whole subcritical regime, we have a weak Poincaré inequality and corresponding polynomial upper bound for the relaxation of the associated Glauber dynamics.

math.PR↗
arXiv2011

Concentration bounds for entropy estimation of one-dimensional Gibbs measures

We obtain bounds on fluctuations of two entropy estimators for a class of one-dimensional Gibbs measures on the full shift. They are the consequence of a general exponential inequality for Lipschitz functions of n variables. The first estimator is based on empirical frequencies of blocks scaling logarithmically with the sample length. The second one is based on the first appearance of blocks within typical samples.

math.DS↗
arXiv2010

Concentration inequalities for Markov processes via coupling

We obtain moment and Gaussian bounds for general Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order 1+epsilon of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order 1+epsilon is finite uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur.

math.PR↗
arXiv2010

On the zero-temperature limit of Gibbs states

We exhibit Lipschitz (and hence Hölder) potentials on the full shift $\{0,1\}^{\mathbb{N}}$ such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space $\{0,1\}^{\mathbb Z}$ for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space $\{0,1\}^{\mathbb{Z}^{d}}$, $d\geq3$, we show that this non-convergence behavior can occur for finite-range interactions, that is, for locally constant potentials.

math-ph↗
arXiv2008

Almost Sure Central Limit Theorems and the Erdos-Renyi law for Expanding Maps of the Interval

For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost sure central limit theorem (ASCLT). In fact, we provide a speed of convergence in the Kantorovich metric. Maxima of partial sums are also shown to obey an ASCLT. The key-tool is an exponential inequality recently obtained. Then we derive almost-sure convergence rates for the supremum of moving averages of Lipschitz observables (Erdos-Renyi type law). We end up with an application to entropy estimation ASCLT's that refi ne Shannon-McMillan-Breiman and Ornstein-Weiss theorems.

math.PR↗
arXiv2008

Poisson processes for subsystems of finite type in symbolic dynamics

Let $Δ\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an irreducible and aperiodic shift of finite type $(Σ_{A}^{+},§)$. Let $Σ_Δ$ be the subshift of allowable paths in the graph of $Σ_{A}^{+}$ which only passes through the vertices of $Δ$. For a random point $x$ chosen with respect to an equilibrium state $μ$ of a Hölder potential $ϕ$ on $Σ_{A}^{+}$, let $τ_{n}$ be the point process defined as the sum of Dirac point masses at the times $k>0$, suitably rescaled, for which the first $n$-symbols of $§^k x$ belong to $Δ$. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of $ϕ$ to $Σ_Δ$ and the parameters of the limit law are explicitly computed.

math.DS↗
arXiv2008

On the asymptotic measure of periodic subsystems of finite type in symbolic dynamics

Let $Δ\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an aperiodic shift of finite type $(Σ_{A}^{+},§)$. Let $Δ_{n}$ be the union of cylinders in $Σ_{A}^{+}$ corresponding to the points $x$ for which the first $n$-symbols of $x$ belong to $Δ$ and let $μ$ be an equilibrium state of a Hölder potential $ϕ$ on $Σ_{A}^{+}$. We know that $μ(Δ_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $μ(Δ_{n})$ and compare it with the pressure of the restriction of $ϕ$ to $Σ_Δ$. The present paper extends some results in \cite{CCC} to the case when $Σ_Δ$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.

math.DS↗
arXiv2008

A concentration inequality for interval maps with an indifferent fixed point

For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^n\to\R$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.

math.DS↗