Numerical simulation of Alex Barnett have shown that nodal sets of large degree $$extract_itex$$N$$/extract_itex$$ random wave on the 3-dimensional space are very different from those on the 2-dimensional space: only one giant component shows up in the graphics (although Nazarov-Sodin show that there are increasing number of components as degree tends to $$extract_itex$$+\infty$$/extract_itex$$). P. Sarnak posed the problem of computing the expected genus of the giant component and proposed that it has maximal order $$extract_itex$$N^3$$/extract_itex$$. Together with S. Zelditch, I prove that these properties hold for real and imaginary parts of random equivariant spherical harmonics of degree $$extract_itex$$N$$/extract_itex$$. This is joint work with S. Zelditch.