Exact roughness exponent of KPZ surfaces in 2+1 dimensions
In plain words
For a growing two-dimensional surface, height fluctuations grow with system size as a power law with exponent $\alpha$. Simulations give $\alpha$ near 0.39, but no theory derives it.
Precise statement
For the KPZ equation $dh/dt = \nu \mathrm{lap}\ h + (\lambda/2)(\mathrm{grad}\ h)^2 + \eta(x,t)$ with Gaussian white noise $\eta$ in $d = 2$ spatial dimensions, determine the roughness exponent $\alpha$ (with $z = 2 - \alpha$ from Galilean invariance) exactly or to $10^{-4}$, and decide whether it is a simple rational number. Large simulations give $\alpha$ approximately 0.387 to 0.39; the conjecture $\alpha = 2/5$ is disfavored.
What would settle it
An exact solution, or a nonperturbative RG computation with controlled error, agreeing with simulations to $10^{-4}$.
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