MARK KAC SEMINAR

I describe the effect of synchronisation of dynamical systems by common noisy force. The main criterion is negative largest Lyapunov exponent of the dynamics. This exponent is generally negative for small noise, but can become positive if the noise is large. In the case of noisy phase oscillators, an extended description for states not close to synchrony can be formulated. An interesting effect appears when common noise competes with repulsive coupling between the oscillators. In this case usual characterisations of synchrony, phase locking and frequency entrainment, appear to be contradictory: one observes phase locking and frequency anti-entrainment.

For many types of random planar maps, i.e. planar graphs embedded in the sphere, it is known that their geometry possesses a scaling limit described by a universal random continuous metric space known as the Brownian sphere. I will give a short overview of its properties and its relation to Liouville Quantum Gravity and discuss ways to escape the universality class by introducing matter degrees of freedom or vertices with heavy-tailed degree distribution. The peeling process will be introduced as a general method to explore planar maps, and can be used to study the geometry of models in the Brownian sphere universality class and beyond.