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.

Date

Speakers

Ivan Corwin

Affiliation

Clay Mathematics Institute, Columbia University and MIT