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Title:
KPZ line ensemble
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Abstract:
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.
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