## Probability Seminar - Jamie Mingo (Queen's University)

### Tuesday, February 13th, 2018

**Time:** 3:30-5:00 p.m. **Place:** Jeffery Hall 319

**Speaker:** Jamie Mingo (Queen's University)

**Title:** The Infinitesimal Law of the GOE, Part II

**Abstract: ** If $X_N$ is the $N \times N$ Gaussian Orthogonal Enemble (GOE) of random matrices, we can expand $\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in $1/N$, often called a genus expansion. Following the celebrated formula of Harer and Zagier for the GUE, Ledoux (2009) found a five term recurrence for the coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show that the coefficient of $1/N$ counts the number of non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.