# Mathematics Colloquia and Seminars

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### Airy process with wanderers, KPZ, and random matrices

**Mathematical Physics & Probability**

Speaker: | Karl Liechty, DePaul University |

Related Webpage: | https://math.depaul.edu/kliechty/ |

Location: | Zoom lecture |

Start time: | Wed, Dec 2 2020, 4:10PM |

In random matrix theory the Tracy-Widom GOE distribution describes the location of the largest eigenvalue of large real symmetric random matrices. It also describes height fluctuations of certain random growth models in the KPZ universality class, and can be characterized as the maximum value of the Airy process minus a parabola. This characterization has a natural deformation when the Airy process is replaced with the Airy process with wanderers, a process whose marginal distributions correspond to spiked complex Hermitian random matrix models like GUE.

I will characterize the distribution of this maximal height in several ways: with a Fredholm determinant formula; in terms of PainlevĂ© transcendents; as a KPZ fixed point distribution with a particular initial condition; and as a solution to two PDEs, the KdV equation and the Bloemendal-Virag equation for spiked random matrices. This is joint work with Daniel Remenik and Gia Bao Nguyen.