Quantized Scaling of Growing Surfaces

Abstract

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 $χ= 2/5, z = 8/5$ for two-dimensional and $χ= 2/7, z = 12/7$ for three-dimensional surfaces are derived.