Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scale-free percolation

Maria Deijfen, Remco van der Hofstad, and Gerard Hooghiemstra

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We formulate and study a model for inhomogeneous long-range percolation on $\mathbb{Z}^{d}$. Each vertex $x\in\mathbb{Z}^{d}$ is assigned a non-negative weight $W_{x}$, where $(W_{x})_{x\in\mathbb{Z}}^{d}$ are i.i.d. random variables. Conditionally on the weights, and given two parameters $\alpha,\lambda>0$, the edges are independent and the probability that there is an edge between $x$ and $y$ is given by $p_{xy}=1-\exp\{-\lambda W_{x}W_{y}/|x-y|^{\alpha}\}$. The parameter $\lambda$ is the percolation parameter, while $\alpha$ describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices.

First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of $W_{x}$ is regularly varying with exponent $\tau-1$, then the tail of the degree distribution is regularly varying with exponent $\gamma=\alpha(\tau-1)/d$. The parameter $\gamma$ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and $\gamma$ are formulated for the existence of a critical value $\lambda_{\mathrm{c}}\in(0,\infty)$ such that the graph contains an infinite component when $\lambda>\lambda_{\mathrm{c}}$ and no infinite component when $\lambda<\lambda_{\mathrm{c}}$. Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point $\gamma=2$, that is, at the point where the degrees switch from having finite to infinite second moment.

The model can be viewed as an interpolation between long-range percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.


Nous définissons et étudions un modèle de percolation inhomogènes à longue portée sur $\mathbb{Z}^{d}$. A chaque site $x\in\mathbb{Z}^{d}$ est assigné un poids positif $W_{x}$, où les $(W_{x})_{x\in\mathbb{Z}^{d}}$ sont des variables aléatoires indépendantes et identiquement distribuées. Conditionnellement aux poids et étant donnés deux paramètres $\alpha,\lambda>0$, les arêtes sont indépendantes et la probabilité qu’il existe un lien entre $x$ et $y$ est $p_{xy}=1-\exp\{-\lambda W_{x}W_{y}/|x-y|^{\alpha}\}$. Le paramètre $\lambda$ est le paramètre de percolation tandis que $\alpha$ caractérise la portée des interactions. Nous étudierons la distribution des degrés dans le graphe résultant et l’existence éventuelle d’une composante infinie ainsi que la distance de graphe entre deux sites éloignés.

Nous montrons d’abord que la queue de la distribution des degrés est liée à la queue de la distribution des poids. Quand la queue de la distribution de $W_{x}$ est à variation régulière d’indice $\tau-1$, alors la queue de la distribution des degrés est à variation régulière d’indice $\gamma=\alpha(\tau-1)/d$. Le paramètre $\gamma$ s’avère crucial pour décrire le modèle. Des conditions sur la distribution des poids et de $\gamma$ sont formulées pour l’existence d’une valeur critique $\lambda_{\mathrm{c}}\in(0,\infty)$ telle que le graphe contienne une composante infinie quand $\lambda>\lambda_{\mathrm{c}}$ et aucune composante infinie quand $\lambda<\lambda_{\mathrm{c}}$. De plus, une transition de phase est établie pour la distance dans le graphe de la composante infinie au point $\gamma=2$, c’est à dire au point où les degrés n’ont plus de second moment fini.

Notre modèle peut être vu comme une interpolation entre la percolation à longue portée et des modèles de graphes aléatoires inhomogènes. Nous montrons qu’il possède les caractéristiques des deux modèles précédents.

Article information

Ann. Inst. H. Poincaré Probab. Statist. Volume 49, Number 3 (2013), 817-838.

First available in Project Euclid: 2 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]

Random graphs Long-range percolation Percolation in random environment Degree distribution Phase transition Chemical distance Graph distance


Deijfen, Maria; van der Hofstad, Remco; Hooghiemstra, Gerard. Scale-free percolation. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 3, 817--838. doi:10.1214/12-AIHP480.

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