First-passage percolation on random geometric graphs: Asymptotic shape and further results

Abstract

In recent years, the study of random geometric graphs, also known as the Gilbert disk model, and their properties has captured significant attention within percolation theory. This presentation delves into the first-passage percolation model, employing independent and identically distributed random variables on the infinite connected component of a random geometric graph. We establish sufficient conditions for the existence of the asymptotic shape, demonstrating that it converges to an Euclidean ball. Furthermore, we will discuss ongoing research on the speed of convergence, along with insights into the bounds governing the fluctuations of geodesic paths.

This talk is based on collaborative works with C.F. Coletti, D. Valesin, A. Hinsen, and B. Jahnel.

About Lucas

Lucas R. de Lima is a PhD candidate at the Federal University of ABC (UFABC) in Brazil, specializing in probability, interacting systems, and percolation. He holds degrees in Mathematics and Science & Technology, with exchange experience at the Freie Universität Berlin, and has served as a guest researcher at the University of Groningen.