We shall revisit some phase transitions in high-dimensional multiple testing problems under sparsity assumptions, and then proceed to characterize some new ones that we recently discovered. In particular, I will describe the signal sizes necessary and sufficient for statistical procedures to simultaneously control false discovery (in terms of family-wise error rate or false discovery rate) and missed detection (in terms of family-wise non-discovery rate or false non-discovery rate) in the simple but ubiquitous signal-plus-noise model

x(i) = \mu(i) + \epsilon(i), \quad i=1,2,\ldots,p

Several well-known procedures are shown to attain said boundaries. Remarkably, these phase transition phenomena continue to hold under a much wider class of models, and under extremely weak dependence assumptions. We provide point-wise, rather than minimax, results, wherever we can. Important practical implications, along with an interesting manifestation of the phase transitions in genome-wide association studies (GWAS), will be discussed.

Behind the statistical results is a probabilistic phenomenon known as relative stability. Much like how the law of large numbers describes the concentration of averages, relative stability --- or the "law of large dimensions" --- describes the concentration of maxima. We provide a complete characterization of the relative stability phenomenon for Gaussian triangular arrays in terms of their correlation structure. Its proof uses classic Sudakov-Fernique and Slepian lemma arguments along with a curious application of Ramsey's coloring theorem.