This paper introduces the largest connected subgraph game played on an undirected graph G. In each round, Alice first colours uncoloured vertex of G red, and then, Bob blue, with all vertices initially uncoloured. Once are coloured, (Bob, resp.) wins if there is a red (blue, whose order greater than any blue (red, subgraph. We prove that, plays optimally, then can never win, define large class ...

In these notes, we survey developments on the asymptotic behavior of the largest eigenvalues of random matrix and random growth models, and describe the corresponding known non-asymptotic exponential bounds. We then discuss some elementary and accessible tools from measure concentration and functional analysis to reach some of these quantitative inequalities at the correct small deviation rate ...