Search

Consider a critical Erdos–Rényi random graph: $n$ is the number of vertices, each one of the $\tbinom{n}{2}$ possible edges is kept in the graph independently from the others with probability $n^{-1}+\lambda n^{-4/3}$, $\lambda $ being a fixed real number. When $n$ goes to infinity, Addario-Berry, Broutin and Goldschmidt (Probab. Theory Related Fields 152 (2012) 367–406) have shown that the collection of connected components, viewed as suitably normalized measured compact metric spaces, converges in distribution to a continuous limit $\mathcal{G}_{\lambda }$ made of random real graphs. In this paper, we consider notably the dynamical percolation on critical Erdos–Rényi random graphs. To each pair of vertices is attached a Poisson process of intensity $n^{-1/3}$, and every time it rings, one resamples the corresponding edge. Under this process, the collection of connected components undergoes coalescence and fragmentation. We prove that this process converges in distribution, as $n$ goes to infinity, toward a fragmentation-coalescence process on the continuous limit $\mathcal{G}_{\lambda }$. We also prove convergence of discrete coalescence and fragmentation processes and provide general Feller-type properties associated to fragmentation and coalescence.

Nous considerons le modele de percolation dynamique sur le graphe complet $K_{n}$, ou chaque arete reactualise son etat au taux $\mu \ll 1/n$, et est ensuite declaree ouverte avec probabilite $p=\lambda /n$, ou $\lambda >1$. Nous etudions une marche aleatoire sur cet environnement dynamique qui saute a taux $1/n$ a travers chaque arete ouverte. Nous montrons que le temps de melange de tout ce processus a un cutoff au temps $\frac{3}{2}\log n/\mu $. Nous l’obtenons en montrant que la composante marche aleatoire melange plus vite que le processus d’environnement; au passage nous controlons le temps que met la marche avant d’etre isolee.

We consider the model of random walk on dynamical percolation introduced by Peres, Stauffer and Steif (2015). We obtain comparison results for this model for hitting and mixing times and for the spectral-gap and log-Sobolev constant with the corresponding quantities for simple random walk on the underlying graph $G$, for general graphs. When $G$ is the torus $\mathbb{Z}_n^d$, we recover the results of Peres et al. and we also extend them to the critical case. We also obtain bounds in the cases where $G$ is a transitive graph of moderate growth and also when it is the hypercube., Comment: 40 pages. Submitted. This is a revised version of the previously titled "Random walk on dynamical percolation". Section 2 was substantially extended

The background subsection in the introduction was significantly expanded.The organization has changed, the main results are highlighted in section 3, some technical results are postponed in section 4 in order to have the main results stated earlier and other technical results are postponed to an appendix. Furthermore, a section dedicated to the perspectives has been added.Some important mistakes have been corrected:- the inappropriate use of compact convergence for stochastic processes, which causes measurability issues. Now, the usual Skorokhod topology is used (cf. appendix A).- the fact that even if a m.s-m.s is a collection of real graphs in $\mathcal{S}$, it is not necessarily the case that after a positive time, its coalescent remains a graph (for instance, one component might have infinite surplus), whence the introduction of the class $\mathcal{S}^{length}$ in section 2.6. There was thus a mistake in Lemma 4.10 (iii) (old numbering), corrected now in Lemma 5.9 (new numbering). See also the remarks after Theorem 3.3.- there was an error, now fixed, concerning the control of supdiam in section 5.4.- Lemma 2.5 (old numbering) as stated was false, a corrected version replaces it.; Consider a critical Erdös-Rényi random graph: $n$ is the number of vertices, each one of the $\binom{n}{2}$ possible edges is kept in the graph independently from the others with probability $n^{−1}$ + $\lambda n^{−4/3}$ , $\lambda$ being a fixed real number. When $n$ goes to infinity, Addario-Berry, Broutin and Goldschmidt have shown that the collection of connected components, viewed as suitably normalized measured compact metric spaces, converges in distribution to a continuous limit $G_\lambda$ made of random real graphs. In this paper, we consider notably the dynamical percolation on critical Erdös-Rényi random graphs. To each pair of vertices is attached a Poisson process of intensity $n^{−1/3}$, and every time it rings, one resamples the corresponding edge. Under this process, the collection of connected components undergoes coalescence and fragmentation. We prove that this process converges in distribution, as $n$ goes to infinity, towards a fragmentation-coalescence process on the continuous limit $G_\lambda$. We also prove convergence of discrete coalescence and fragmentation processes and provide Feller-type properties associated to fragmentation and coalescence.

We study the contact process running in the one-dimensional lattice undergoing dynamical percolation, where edges open at rate $vp$ and close at rate $v(1-p)$. Our goal is to explore how the speed of the environment, $v$, affects the behavior of the process. We show in particular that for small enough $v$ the process dies out, while for large $v$ the process behaves like a contact process on $\mathbb{Z}$ with rate $\lambda p$, so it survives if $\lambda$ is large. We also show that if $v$ and $p$ are small then the network becomes immune, in the sense that the process dies out for any infection rate $\lambda$, while if $p$ is sufficiently close to $1$ then for all $v>0$ survival is possible for large enough $\lambda$., Comment: 14 pages, 4 figures. New results were added about the extinction time of the process and about extensions to general vertex-transitive graphs. To appear in the Electronic Journal of Probability

We study three models of percolation in the plane: Bernoulli percolation, Voronoi percolation, and nodal lines percolation. Bernoulli percolation is often considered as the simplest model which admits a phase transition. Voronoi percolation is a Bernoulli percolation model in random environment. Nodal lines percolation is a level lines percolation model for smooth planar Gaussian fields. We have followed two main threads. The first one is the resarch of similarities between these models, having in mind that we expect that they admit the same scaling limit. We show for instance that the critical level for nodal lines percolation is the self-dual level (namely the zero level) if the Gaussian field is the Bargmann-Fock field, which is natural analytical field. The second main thread is the study of dynamics on these percolation models. We show in particular that if we sample a critical Voronoi percolation model and if we let each point move according to a long range stable Lévy process, then there exist exceptional times with an unbounded cluster, Dans cette thèse, nous étudions trois modèles de percolation planaire : la percolation de Bernoulli, la percolation de Voronoi, et la percolation de lignes nodales. La percolation de Bernoulli est souvent considérée comme le modèle le plus simple à définir admettant une transition de phase. La percolation de Voronoi est quant à elle un modèle de percolation de Bernoulli en environnement aléatoire. La percolation de lignes nodales est un modèle de percolation de lignes de niveaux de champs gaussiens lisses. Deux fils conducteurs principaux ont guidé nos travaux. Le premier est la recherche de similarités entre ces modèles, en ayant à l'esprit que l'on s'attend à ce qu'ils admettent tous la même limite d'échelle. Nous montrons par exemple que le niveau critique de la percolation de lignes nodales est égal au niveau auto-dual (à savoir le niveau zéro) lorsque le champ considéré est le champ de Bargmann-Fock, qui est un champ gaussien analytique naturel. Le deuxième fil conducteur est l'étude de dynamiques sur ces modèles. Nous montrons en particulier que, si on considère un modèle de percolation de Voronoi critique et si on laisse les points se déplacer selon des processus de Lévy stables à très longue portée, alors il existe des temps exceptionnels avec une composante non bornée

We consider dynamical percolation on the d-dimensional discrete torus Znd of side length n, where each edge refreshes its status at rate μ=μn≤1/2 to be open with probability p. We study random walk on the torus, where the walker moves at rate 1 / (2d) along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case ppc(Zd), the mixing time is Θ(n2+1/μ); here the implied constants depend only on d and p. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption θ(p)>1/2. When θ(p)>0, we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times. The latter implies sharp (up to poly-logarithmic factors) upper bounds on exit times of large balls throughout the supercritical regime. Our proofs are based on percolation results (e.g., the Grimmett–Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time.

In dynamical critical site percolation on the triangular lattice or bond percolation on \Z^2, we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten's Incipient Infinite Cluster. In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is different. We also study the collapse of the infinite cluster near typical exceptional times, and establish a relation between static and dynamic exponents, analogous to Kesten's near-critical relation., 56 pages, 7 figures. In this version, we prove convergence only for one of our local time constructions. Consequently, some arguments are slightly changed in Sections 2 and 3

This thesis combines the study of asymptotic properties of percolation processes with various dynamical concepts. First-passage percolation is a model for the spatial propagation of a fluid on a discrete structure; the Shape Theorem describes its almost sure convergence towards an asymptotic shape, when considered on the square (or cubic) lattice. Asking how percolation structures are affected by simple dynamics or small perturbations presents a dynamical aspect. Such questions were previously studied for discrete processes; here, sensitivity to noise is studied in continuum percolation. Paper I studies first-passage percolation on certain 1-dimensional graphs. It is found that when identifying a suitable renewal sequence, its asymptotic behaviour is much better understood compared to higher dimensional cases. Several analogues of classical 1-dimensional limit theorems are derived. Paper II is dedicated to the Shape Theorem itself. It is shown that the convergence, apart from holding almost surely and in L^1, also holds completely. In addition, inspired by dynamical percolation and dynamical versions of classical limit theorems, the almost sure convergence is proved to be dynamically stable. Finally, a third generalization of the Shape Theorem shows that the above conclusions also hold for first-passage percolation on certain cone-like subgraphs of the lattice. Paper III proves that percolation crossings in the Poisson Boolean model, also known as the Gilbert disc model, are noise sensitive. The approach taken generalizes a method introduced by Benjamini, Kalai and Schramm. A key ingredient in the argument is an extremal result on arbitrary hypergraphs, which is used to show that almost no information about the critical process is obtained when conditioning on a denser Poisson process.