The Kardar-Parisi-Zhang exponents for the $2+1$ dimensions

The Kardar-Parisi-Zhang (KPZ) equation has been connected to a large number of important stochastic processes in physics, chemistry and growth phenomena, ranging from classical to quantum physics. The central quest in this field is the search for ever more precise universal growth exponents. Notably, exact growth exponents are only known for $1+1$ dimensions. In this work, we present physical and geometric analytical methods that directly associate these exponents to the fractal dimension of the rough interface. Based on this, we determine the growth exponents for the $2+1$ dimensions, which are in agreement with the results of thin films experiments and precise simulations. We also make a first step towards a solution in $d+1$ dimensions, where our results suggest the inexistence of an upper critical dimension.