Fractal Weyl laws in discrete models of chaotic scattering

We analyse simple models of quantum chaotic scattering, namely quantized open baker's maps. We numerically compute the density of quantum resonances in the semiclassical regime. This density satisfies a fractal Weyl law, where the exponent is governed by the (fractal) dimension of the set of trapped trajectories. This type of behaviour is also expected in the (physically more relevant) case of Hamiltonian chaotic scattering. Within a simplified model, we are able to rigorously prove this Weyl law and compute quantities related to the 'coherent transport' through the system, namely the conductance and 'shot noise'. The latter is close to the prediction of random matrix theory.