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

When |
Jun 20, 2016
from 11:00 AM to 12:00 PM |
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Where | Amphi. Schrödinger |

Attendees |
Satya Majumdar |

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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.