Quantized Scaling of Growing Surfaces

The Kardar-Parisi-Zhang universality class of stochastic surface growth is studied by exact field-theoretic methods. From previous numerical results, a few qualitative assumptions are inferred. In particular, height correlations should satisfy an operator product expansion and, unlike the correlations in a turbulent fluid, exhibit no multiscaling. These properties impose a quantization condition on the roughness exponent χ and the dynamic exponent $z$. Hence the exact values $\chi =2/5,z=8/5\mathrm{}$ for two-dimensional and $\chi =2/7,z=12/7\mathrm{}$ for three-dimensional surfaces are derived.