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

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

2010-02-09

arxiv.org/abs/1002.1883v3

Phys. Rev. Lett. 104, 230602 (2010)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 2 figures, revised

Scientific paper

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. Within this regime the KPZ equation details how the long time asymptotics is approached.

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