Topology of the nodal set of random equivariant spherical harmonics on $S^3$

Abstract

We show that real and imaginary parts of equivariant spherical harmonics on $S^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m\left(\frac{N^2-m^2}{2}+N\right)$. Hence if $\frac{m}{N}=c$ for fixed $0<c<1$, the genus has order $N^3$.

arXiv comment15 pages

References in corpus (2)

• Critical points and supersymmetric vacua
• Universality and scaling of zeros on symplectic manifolds