KPZ line ensemble

Ivan Corwin
Massachusetts Institute of Technology

We construct a 'line ensemble' extension for the solution to the narrow wedge initial data KPZ equation and prove that it displays a sort of spatial Markov property we call the 'H-Brownian Gibbs property'. This line ensemble and its properties provide vital regularity information about the KPZ equation (even when scaled horizontally by t^{2/3} and vertically by t^{1/3}). We show four applications of our result:
(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.

Presentation (PDF File)

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