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Monday programme

Room B3.03

Morning Session Chair: Gernot Akemann

Afternoon Session Chair: Mario Kieburg

Time Speaker Title Abstract
10:00-10:55 Coffee in the Common Room
10:55-11:00 Welcome from organisers
11:00-11:50 Arno Kuijlaars Propagation of singular behavior in UE and GUE sums

I will discuss sums of Hermitian random matrices $M + \sqrt{\tau} H$
where $M$ is a matrix from a unitary ensemble of the form
$$ \frac{1}{Z_n} e^{-n Tr V(M)}$$ and $H$ is a scaled GUE matrix.
For very special choices of the potential $V$ the eigenvalues of $M$ has
a density that vanishes at an interior point, or vanishes to higher
order at an edge point. We show that the singular behavior persists for the eigenvalues of $M + \sqrt{\tau} H$ for $\tau$ up to a critical value $\tau_{cr}$.
In addition, the local scaling limits at the singular point continue to hold as
well. This is joint work with Tom Claeys, Karl Liechty and Dong Wang

11:50-12:40 Pierpaolo Vivo Universal fluctuation formulae for one-cut β-ensembles - with a combinatorial touch I discuss a recently obtained analytical formula for the covariance Cov(A,B) of two smooth linear statistics on the eigenvalues of a one-cut β-ensemble of random matrices. This allows to compute the generating function of the covariances of power traces for one-cut β-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. I briefly discuss the connection with the combinatorial problem of non-crossing pairings of 2 circles with ‘k’ and ‘l’ points on them, respectively.

In collaboration with Francesco Mezzadri and Fabio Deelan Cunden

[J. Phys. A: Math. Theor. 48, 315204 (2015); Phys. Rev. Lett. 113, 070202 (2014)]

12:40-14:10 Lunch in the Common Room
14:10-15:00 Khanh Duy Trinh On spectral measures of beta ensembles

This talk concerns with the limiting behaviour of spectral measures of random Jacobi matrices of Gaussian, Wishart and MANOVA beta ensembles. We show that the spectral measures converge weakly to a limit distribution which is the semicircle distribution, Marchenko-Pastur distributions or the arcsine distribution, respectively. Regarding that
convergence as the law of large number, a central limit theorem is then derived.

15:00-16:00 Tea in the Common Room
16:00-16:50 Chris Joyner  

The probability distribution of spectral moments for the Gaussian beta-ensembles

We derive the joint probability distribution of the first two spectral moments for the Gaussian beta-Ensembles random matrix ensembles in N dimensions for any N. This is achieved by mak-ing use of two complementary invariants of the domain where the spectral moments are de-fined. Our approach is significantly different from those employed previously to answer related questions and potentially offers new insights. We also discuss the problems faced when at-tempting to include higher spectral moments. This is work together with U. Smilansky and T. Maciazek
18:00-20:00 Dinner in the Common Room