Brownian structure in the KPZ fixed point

Understand

Many models of one-dimensional local random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class.

• For such a model, the interface profile at advanced time may be viewed in scaled coordinates specified via characteristic KPZ scaling exponents of one-third and two-thirds.
• When the long time limit of this scaled interface is taken, it is expected -- and proved for a few integrable models -- that, up to a parabolic shift, the Airy$_2$ process $\mathcal{A}:\mathbb{R} \to \mathbb{R}$ is obtained.
• This process may be embedded via the Robinson-Schensted-Knuth correspondence as the uppermost curve in an $\mathbb{N}$-indexed system of random continuous curves, the Airy line ensemble.