Tracy-Widom distribution, Airy2 process and its sample path properties

Abstract

Tracy-Widom distribution was first discovered in the study of largest eigenvalues of high dimensional Gaussian unitary ensembles (GUE), and since then it has appeared in a number of apparently distinct research fields. It is believed that Tracy-Widom distribution have a universal feature like classic normal distribution. Airy2 process is defined through finite dimensional distributions with Tracy-Widom distribution as its marginal distributions. In this introductory survey, we will briefly review some basic notions, intuitive background and fundamental properties concerning Tracy-Widom distribution and Airy2 process. For sake of reading, the paper starts with some simple and well-known facts about normal distributions, Gaussian processes and their sample path properties.