Asymptotic Shape of Subadditive Processes on Groups and on Random Geometric Graphs
This doctoral thesis undertakes an in-depth exploration of limiting shape theorems across
diverse mathematical structures, with a specific focus on subadditive processes within
finitelygeneratedgroupsexhibitingpolynomialgrowthrates,aswellasstandard
First-Passage Percolation (FPP) models applied to Random Geometric Graphs (RGGs).
Employing a diverse range of techniques, including subadditive ergodic theorems and
tailored modifications suited for polygonal paths within groups, the thesis examines the
asymptotic shape under varying conditions.The investigation extends to subadditive
cocycles characterized by at least and at most linear growth. Moreover, the study delves
into moderate deviations for FPP models on RGGs, refining previous results with theorems
that quantify its speed of convergence to the limiting shape, the fluctuation of the
geodesics, and its spanning trees.Additionally, we apply the obtained results in a
competition model to verify the positive probability of coexistence of two species
competing for territory in a random geometric graph.