Non-equilibrium Dynamics and Random Matrices

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.