PCA, likelihood ratios, and a transition to the Tracy-Widom law

The Tracy-Widom distribution has found broad use in statistical theory and application. We review some of this, focusing first on Principal Components Analysis and the `spiked model'. When the spike signal is below the Baik-Ben Arous-Peche threshold, likelihood ratio tests for presence of a spike are more efficient than the largest eigenvalue in many settings of multivariate statistics. In recent work with Egor Klochkov, Alexei Onatski and Damian Pavlyshyn, we study the likelihood ratio test in the transition zone around the BBP threshold, making use of a connection with the spherical Sherrington-Kirkpatrick model.