Exponential rate for the contact process extinction time

Schapira, Bruno; Valesin, Daniel

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 3, pp. 503-526.

Nous étudions le temps d’extinction du processus de contact sur des suites de graphes finis, issus de familles de graphes aléatoires classiques. Sous l’hypothèse d’un taux d’infection supérieur à sa valeur critique sur $\mathbb{Z}$, nous montrons que le logarithme du temps d’extinction divisé par la taille du graphe converge en probabilité vers une constante positive (dépendant du modèle considéré). La famille de graphes considérés inclut divers modèles de percolation, en régime surcritique, sur des sous-boîtes croissantes de ${\mathbb{Z}}^d$ ou ${ℝ}^d$ (percolation de Bernoulli par site ou par arête, ensemble des entrelacs aléatoires et son complémentaire, ensemble d’excursions du champ libre gaussien, graphe aléatoire géométrique), ainsi que les arbres de Galton–Watson surcritiques tronqués à une hauteur finie.

We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of ${\mathbb{Z}}^d$ or ${ℝ}^d$ in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton–Watson trees grown up to finite generations.

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