Universality of the Airy processes along space-like paths

Half a decade ago, Praehofer and Spohn discovered the Airy_2 process corner growth model. It appeared to be one of the universal processes, appearing in different models including random matrix theory. More recently, we discovered the analogue for growth on a flat substrate: the Airy_1 process. This process does not describe only the large time surface statistics at a fixed time, but its universality extends to any "space-like paths"! We will present the result using the totally asymmetric simple exclusion process (TASEP) as reference model, for which the extreme special cases of space-like paths are (a) fixed time, and (b) tagged particle.