How big are the increments of Airy lines?

Airy line ensemble is a sequence of real-valued random processes, whose finite-dimensional distributions are determined by the determinantal of the so-called generalized Airy kernel. This line ensemble was first introduced and studied by Prähofer and Spohn in the study of random, growth models (polynuclear growth model) in 2002, and the later research work shows that the Airy line ensemble can be used as limit curves of other types of random processes. The mathematically rigorous proof of its existence was given by Corwin and Hammond in 2014. Compared with Brownian motion and Gaussian processes, the Airy line ensemble is a new type of random process, but it is seen as an interesting and useful, and even a challenging object through recent works, thereby worthy of further attention. Based on personal research interest, we briefly survey some recent advances in the study of properties of sample paths, including continuity, the modulus of continuity (increments), and the distribution of extremes.