Top eigenvalue of a random matrix: Tracy-Widom distribution and third order phase transition – lecture 1

Tracy-Widom distribution describes the probability distribution of the typical fluctuations of the top eigenvalue of a Gaussian $(NxN)$ random matrix. Over the last decade, the same distribution has surfaced in a wide variety of problems from Kardar-Parisi-Zhang (KPZ) surface growth, directed polymer, random permutations, all the way to large $N-$gauge theory and wireless communications, with some of these problems having no apriori connection to random matrices. Why is the Tracy-Widom distribution so ubiquitous? In statistical physics, universality is usually accompanied by a phase transition–near a critical point often the details become completely irrelevant. So, is there an underlying phase transition associated with the Tracy-Widom distribution?
In this talk, I will demonstrate that for large but finite $N$, indeed there is an underlying third order phase transition from a “strong” coupling to a “weak” coupling phase–the Tracy-Widom distribution turns out to be the universal crossover function between these two phases for finite but large N. Several examples of this third order phase transition will be discussed.