Local Behaviour of Airy Processes

The Airy processes describe limit fluctuations in a wide range of growth models, where each particular Airy process depends on the geometry of the initial profile. We show how the coupling method, developed in the last-passage percolation context, can be used to prove existence of a continuous version and local convergence to Brownian motion. By using similar arguments, we further extend these results to a two parameter limit fluctuation process (Airy sheet).