## Distribution of the ratio of level spacings in random matrix ensembles.

### Yasar Atas, LPTMS

Initially introduced as a description of energy levels of heavy atomic nuclei by Wigner, Random Matrix Theory (RMT) is nowadays an active field of theoretical physics with ramification in various disciplines such as number theory, quantum chaos and finance to cite just a few. In RMT, the distribution of level spacings plays a very important role: it has been widely used since the inception of the theory and is considered as the « reference » measure of spectral statistics. Since different models may and do have different level densities, one has to perform a procedure on the spectrum called unfolding in order to obtain the spacing distribution. This procedure is not unique and the choice of the unfolding procedure is quite arbitrary. It seems then natural to search for another measure which is independent of the level density. This has been done quite recently by Oganesyan and Huse [1]: instead of looking at the spacing they prefer to look at the ratio of two consecutive level spacings. This quantity has the advantage that it does not require any unfolding.

In this talk, I will make few remarks on random matrix theory and the unfolding procedure. I will then derive simple expressions for the probability distribution of the ratio of two consecutive spacings for the classical ensembles of random matrices [2]. These expressions, which were lacking in the literature, will be compared to numerical data from a quantum many-body lattice model and from zeros of Riemann zeta function.

[1] V. Oganesyan and D. A. Huse, Phys. Rev. B 75, 155111 (2007).

[2] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Phys. Rev. Lett. 110, 084101 (2013)

## Avalanche statistics in disordered visco-elastic interfaces

### François Landes, LPTMS

Many complex systems respond to a continuous input of energy by an accumulation of stress over time, and sudden energy releases, called avalanches. Recently, it has been pointed out that several basic features of avalanche dynamics are induced at the microscopic level by relaxation processes, usually neglected by conventional models. I will present a minimal model with relaxation and its mean field treatment, and a quick snapshot of the finite dimension results. In mean-field, our model yields a periodic behavior (with a new, emerging time scale), with events that span the whole system. In finite dimension (2D), the mean-field system-sized events become local, and numerical simulations give qualitative and quantitative results similar to the earthquakes observed in reality.