KPZ growth in two and more dimensions

5 problems

• Measuring 2+1 KPZ statistics beyond flat thin-film growth

  Partially resolved

  One-dimensional KPZ laws have been seen in liquid-crystal turbulence and in fluids of light, while two-dimensional tests exist only for flat growing films. A clean experiment in another geometry or system would measure the full statistics of a growing two-dimensional surface or its analog, not only its exponents.

• Exact roughness exponent of KPZ surfaces in 2+1 dimensions

  Open

  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.

• Universal height distributions of KPZ surfaces in 2+1 dimensions

  Open

  In one dimension the rescaled height of a KPZ surface follows the Tracy-Widom distributions of random matrix theory (the statistics of the largest eigenvalue of a large random matrix). In two dimensions the corresponding universal distributions are measured numerically but have no known formula.

• Scale invariance at the KPZ roughening transition in three dimensions

  Open

  Above two dimensions a KPZ surface stays smooth when the slope effect is weak and becomes rough when it is strong. A 2021 proposal says the transition point between the two lacks ordinary scale invariance and instead repeats its pattern only at discrete zoom factors.

• Upper critical dimension of the KPZ class

  Contested

  In high enough dimension many fluctuating systems become simple and are described exactly by mean-field theory. It is unknown whether KPZ growth has such a dimension, above which the rough phase loses its nontrivial roughness, or whether nontrivial roughness persists in every dimension.