# On the largest eigenvalue of a random matrix: Tracy-Widom distribution, large deviations and third order phase transition.

#### On the largest eigenvalue of a random matrix: Tracy-Widom distribution, large deviations and third order phase transition.

- Event time: 11:30am until 12:30pm
- Event date: 27th May 2020
- Speaker: Dr Gregory Schehr (Université de Paris-Sud)
- Location: Online - see email.

### Event details

The statistical properties of the largest eigenvalue $\lambda_{\max}$ of a random matrix are of interest in diverse fields such as in the stability of large ecosystems, in disordered systems and related stochastic growth processes, in statistical data analysis and even in string theory. In this talk I will review the developments in the theory of the fluctuations of $\lambda_{\max}$ in the classical Gaussian ensembles of Random Matrix Theory -- such as the Gaussian Orthogonal Ensemble. In the limit of large matrix size $N$, the probability density function (PDF) $\lambda_{\max}$ consists of a central part described by the celebrated Tracy-Widom distribution flanked, on both sides, by two large deviations tails. While the central part characterizes the typical fluctuations of $\lambda_{\max}$, the large deviations tails are instead associated to extremely rare fluctuations. I will discuss in particular the third-order phase transition which separates the left tail from the right tail, a transition akin to the so-called Gross-Witten-Wadia phase transition found in 2-d lattice quantum chromodynamics. If time permits, I will discuss the occurrence of similar third-order transitions in various physical problems, such as the Sardar-Parisi-Zhang equation in $1+1$ dimensions.

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