Maximum degree in minor-closed classes of graphs

Given a class of graphs GG closed under taking minors, we study the maximum degree ΔnΔn of random graphs from GG with nn vertices. We prove several lower and upper bounds that hold with high probability. Among other results, we find classes of graphs providing orders of magnitude for ΔnΔn not observed before, such us logn/logloglognlogn/logloglogn and logn/loglogloglognlogn/loglogloglogn.