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The one-dimensional KPZ equation: an exact solution and its universality · Around

2010

The one-dimensional KPZ equation: an exact solution and its universality

Sasamoto, Tomohiro, Spohn, Herbert

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We report on the first exact solution of the KPZ equation in one dimension, with an initial condition which physically corresponds to the motion of a macroscopically curved height profile.

The solution provides a determinantal formula for the probability distribution function of the height $h(x,t)$ for all $t>0$.
In particular, we show that for large $t$, on the scale $t^{1/3}$, the statistics is given by the Tracy-Widom distribution, known already from the theory of GUE random matrices.
Our solution confirms that the KPZ equation describes the interface motion in the regime of weak driving force.

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