Universal height distributions of KPZ surfaces in 2+1 dimensions
In plain words
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.
Precise statement
For KPZ growth in $d = 2$ with flat, curved (radial) and stationary initial conditions, write $h(x,t)=v t+(\Gamma t)^{\beta}\chi$ and identify the limiting distribution of $\chi$ in each geometry, together with its spatial and temporal correlation functions, and determine whether it has a solvable structure such as a Fredholm determinant or determinantal point process. An answer is an explicit distribution matching simulations.
What would settle it
An exact derivation of the distribution for one solvable 2+1 model, or a closed form that matches high-precision simulations in all moments.