Many models of one dimensional random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For a few such models, the limiting interface profile, after scaling by characteristic KPZ scaling exponents of one-third and two-third, is known to be the Airy_2 process shifted by a parabola. This limiting process is expected to be “locally Brownian”, and a recent result gives a quantified bound on probabilities of events under the Airy_2 process on a unit order interval in terms of probabilities of the same events under Brownian motion (of rate two). This comparison also holds in the prelimit for the particular model of Brownian last passage percolation. In this talk, we will introduce KPZ universality and discuss this result and a number of consequences, using last passage percolation as an expository framework.

Joint work with Jacob Calvert and Alan Hammond.

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Last updated: 02 Jun 2023