KPZ line ensemble

Ivan Corwin, Alan Hammond

Published 2013-12-09, updated 2015-07-13Version 2

For each t>1 we construct an 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 H-Brownian Gibbs property (with 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. have tight Radon-Nikodym derivatives) with respect to Brownian bridges as time t goes to infinity. This construction uses as inputs the diffusion that O'Connell discovered in relation to the O'Connell-Yor semi-discrete Brownian polymer, the convergence result of Moreno Flores-Quastel-Remenik of the lowest indexed curve of that diffusion to the solution of the KPZ equation with narrow wedge initial data, and the one-point distribution formula proved by Amir-Corwin-Quastel for the solution of the KPZ equation with narrow wedge initial data. We provide four main applications of this construction: (1) Uniform (as t goes to infinity) Brownian absolute continuity of the time t solution to the KPZ equation with narrow wedge initial data, even when scaled vertically by t^{1/3} and horizontally by t^{2/3}; (2) Universality of the t^{1/3} one-point (vertical) fluctuation scale for the solution of the KPZ equation with general initial data; (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 solution at fixed time of the KPZ equation with general initial data.