Probability Research for Obtaining Generalized Results on Random Trees and Graphs

My goal is to significantly advance the field of random graphs, and especially random trees, by obtaining general results valid for large classes of random trees and graphs, where others have usually only examined individual cases. To achieve this I will combine a wide range of probabilistic methods (e.g.

Pólya urns, branching processes, fragmentation theory, planar maps, Stein’s method with couplings, the contraction method and renewal theory) with innovative combinatorial approaches.The project has three parts, each with specific aims:A) Percolation and fragmentation processes on large classes of random treesGet sharp results for bond percolation on split trees (a large class of random trees of logarithmic height) as well as on trees with prescribed degrees, the most general class of random trees of non-logarithmic height.Analyze fragmentation processes for both of these major classes of random trees.Investigate bootstrap percolation on Galton Watson (G-W) trees.B) Fringe trees as a novel approach to obtain general results for whole classes of random trees, specifically split trees and conditioned G-W trees.C) The random configuration modelInvestigate the configuration model with heavy-tailed degrees.Develop my new invention of treating the configuration model as an infinite graph sequence for studying the evolution of the process.The studies will significantly add to knowledge in the field with application potentials in e.g. data algorithms and real-world networks.