KPZ line ensemble

Understand

For each $t\geq 1$ we construct an $\mathbf{N}$-indexed ensemble of random continuous curves with three properties: 1.

• The lowest indexed curve is distributed as the time $t$ Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic PDE with narrow wedge initial data; 2.
• The entire ensemble satisfies a resampling invariance which we call the $\mathbf{H}$-Brownian Gibbs property (with $\mathbf{H}(x)=e^{x}$); 3.
• Increments of the lowest indexed curve, when centered by $-t/24$ and scaled down vertically by $t^{1/3}$ and horizontally by $t^{2/3}$, remain uniformly absolutely continuous (i.e.