[AN] Júlia Komjáthy: Large deviations for the giant component in long range percolation.

12 November 2024 16:00 till 17:00 - Location: EEMCS Lecture Hall F | Add to my calendar

Long range percolation is a classical spatial percolation model where long-range interactions are also allowed. The model is typically defined on a lattice like Z^d for dimension d at least 1or on a Poisson point process. Any two vertices x, y are then connected by an edge with probability that is a decays polynomially with the spatial distance. The model is called supercritical if it contains an infinite connected component. This component is known to be almost surely unique. We focus on supercritical models and uncover a connection between three quantities:
1) large deviations for the size of the largest component in a box,
2) the size of the second largest component in a box,
3) the cluster size decay; i.e., the asymptotic behavior of the probability that the origin is a finite component of at least k vertices.
In this talk I will explain the connection between these 3 quantities and show some proof ideas for the case when long edges dominate the behavior.
Joint work with Joost Jorritsma and Dieter Mitsche.