Condensed Matter Seminar: Top eigenvalue of a random matrix: Tracy-Widom distribution and third order phase transition
Satya Majumdar, LPTMS, Université de Paris-Sud
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 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.
Event Organizer: Prof. Eli Eisenberg