KPZ line ensemble

We construct a $\mathrm{KPZ}_t$ line ensemble -- a natural number indexed collection of random continuous curves which satisfies a resampling invariance called the H-Brownian Gibbs property (with $H(x)=e^x$) and whose 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. We prove four main applications of this construction: 1. Uniform (in $t$) Brownian absolute continuity of the fixed time narrow-wedge initial data KPZ equation, even after fluctuation scaling of order $t^{1/3}$ and spatial scaling of order $t^{2/3}$; 2. Universality of the $t^{1/3}$ one-point fluctuation scale for general initial data KPZ equation; 3. Concentration in the $t^{2/3}$ scale for the endpoint of the continuum directed random polymer; 4. Exponential upper and lower tail bounds for the fixed time general initial data KPZ equation. This is joint work with Alan Hammond.