arXiv subjects
Publications and source records attributed to Xavier Bressaud.
The 6-vertex model is a seminal model for many domains in Mathematics and Physics. The sets of configurations of the 6-vertex model can be described as the sets of paths in multigraphs. In this article the transition probability of the simple random walk on the multigraphs is computed. The unexpected point of the results is the use of continuous fractions to compute the transition probability.
We study a two player repeated zero-sum game with asymmetric information introduced by Renault in which the underlying state of the game undergoes Markov evolution (parameterized by a transition probability $\frac 12\le p\le 1$). Hörner, Rosenberg, Solan and Vieille identified an optimal strategy, $σ^*$ for the informed player for $p$ in the range $[\frac 12,\frac 23]$. We extend the range on which $σ^*$ is proved to be optimal to about $[\frac 12,0.719]$ and prove that it fails to be optimal at a value around 0.7328. Our techniques make use of tools from dynamical systems, specifically the notion of pressure, introduced by D. Ruelle.
We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.
This manuscript proposes a mathematical model for the game named Breakout. Its systematic study reveals unexpected complexity. After a survey of general properties of the model, we study in details a special case and exhibit combinatorial difficulties arising in another case.
In this paper, we propose a functional analysis of a set of individual space-speed profiles corresponding to speed as function of the distance traveled by the vehicle from an initial point. This functional analysis begins with a functional modeling of space-speed profiles and the study of mathematical properties of these functions. Then, in a first step, a smoothing procedure based on spline smoothing is developed in order to convert the raw data into functional objets and to filter out the measurement noise as efficiently as possible. It is shown that this smoothing step leads to a complex nonparametric regression problem that needs to take into account two constraints: the use of the derivative information, and a monotonicity constraint. The performance of the proposed two-step estimator (smooth, and then monotonize) is illustrated on simulation studies and a real data example. In a second step, we use a curve registration method based on landmarks alignment in order to construct an average speed profile representative of a set of individual speed profiles. Finally, the variability of such a set is explored by the use of functional boxplots.
This paper presents some optimal real-time and post-processing estimators of vehicle position using odometer and map-matched GPS measurements. These estimators were based on a simple statistical error model of the odometer and the GPS which makes the model generalizable to other applications. Firstly, an asymptotically minimum variance unbiased estimator and two optimal moving fixed interval filters which are more flexibles are exposed. Then, the post-processing case leads to the construction of two moving fixed interval smoothers. These estimators are tested and compared with the classical Kalman filter with simulated and real data, and the results show a good accuracy of each of them.
In this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated to the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition to be a measurable eigenvalue. Then we consider two families of examples. A first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally we study Toeplitz type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove measurable eigenvalues are always rational but not necessarily continuous.
Using an unusual, yet natural invariant measure we show that there exists a sensitive cellular automaton whose perturbations propagate at asymptotically null speed for almost all configurations. More specifically, we prove that Lyapunov Exponents measuring pointwise or average linear speeds of the faster perturbations are equal to zero. We show that this implies the nullity of the measurable entropy. The measure m we consider gives the m-expansiveness property to the automaton. It is constructed with respect to a factor dynamical system based on simple "counter dynamics". As a counterpart, we prove that in the case of positively expansive automata, the perturbations move at positive linear speed over all the configurations.
We consider a family of discrete coagulation-fragmentation equations closely related to the one-dimensional forest-fire model of statistical mechanics: each pair of particles with masses $i,j \in \nn$ merge together at rate 2 to produce a single particle with mass $i+j$, and each particle with mass $i$ breaks into $i$ particles with mass 1 at rate $(i-1)/n$. The (large) parameter $n$ controls the rate of ignition and there is also an acceleration factor (depending on the total number of particles) in front of the coagulation term. We prove that for each $n\in \nn$, such a model has a unique equilibrium state and study in details the asymptotics of this equilibrium as $n\to \infty$: (I) the distribution of the mass of a typical particle goes to the law of the number of leaves of a critical binary Galton-Watson tree, (II) the distribution of the mass of a typical size-biased particle converges, after rescaling, to a limit profile, which we write explicitly in terms of the zeroes of the Airy function and its derivative. We also indicate how to simulate perfectly a typical particle and a size-biased typical particle, which allows us to give some probabilistic interpretations of the above results in terms of pruned Galton-Watson trees and pruned continuum random trees.
Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of modulus 1. We consider the corresponding eigenfunctions, and in Theorem 1.1 we prove that the limit inferior of the ergodic sums is bounded for every point in the phase space. In Theorem 1.2, we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length. Under additional assumptions, we prove that ergodic integrals satisfy the Central Limit Theorem (Theorem 1.3, Theorem 1.9).
We consider the one-dimensional generalized forest fire process: at each site of $\zz$, seeds and matches fall according some i.i.d. stationary renewal processes. When a seed falls on an empty site, a tree grows immediately. When a match falls on an occupied site, a fire starts and destroys immediately the corresponding connected component of occupied sites. Under some quite reasonable assumptions on the renewal processes, we show that when matches become less and less frequent, the process converges, with a correct normalization, to a limit forest fire model. According to the nature of the renewal processes governing seeds, there are four possible limit forest fire models. The four limit processes can be perfectly simulated. This study generalizes consequently a previous result of the authors where seeds and matches were assumed to fall according to Poisson processes.
We consider the so-called one-dimensional forest fire process. At each site of $\mathbb{Z}$, a tree appears at rate $1$. At each site of $\mathbb{Z}$, a fire starts at rate $λ>0$, immediately destroying the whole corresponding connected component of trees. We show that when $λ$ is made to tend to $0$ with an appropriate normalization, the forest fire process tends to a uniquely defined process, the dynamics of which we precisely describe. The normalization consists of accelerating time by a factor $\log(1/λ)$ and of compressing space by a factor $λ\log(1/λ)$. The limit process is quite simple: it can be built using a graphical construction and can be perfectly simulated. Finally, we derive some asymptotic estimates (when $λ\to0$) for the cluster-size distribution of the forest fire process.
We give necessary and sufficient conditions to have measurable and continuous eigenfunctions for linearly recurrent Cantor dynamical systems. We also construct explicitly an example of linearly recurrent system with nontrivial Kronecker factor and a trivial maximal equicontinuous factor.
In this article we prove that given a self-similar interval exchange transformation T, whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of M. Cobo in [C].
We consider an interacting particle system $(η_t)_{t\geq 0}$ with values in $\{0,1\}^{\mathbb{Z}}$, in which each vacant site becomes occupied with rate 1, while each connected component of occupied sites become vacant with rate equal to its size. We show that such a process admits a unique invariant distribution, which is exponentially mixing and can be perfectly simulated. We also prove that for any initial condition, the avalanche process tends to equilibrium exponentially fast, as time increases to infinity. Finally, we consider a related mean-field coagulation-fragmentation model, we compute its invariant distribution, and we show numerically that it is very close to that of the interacting particle system.
We present an upper bound on the mixing rate of the equilibrium state of a dynamical systems defined by the one-sided shift and a non Hölder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.