Let \(\mathcal H_n\) be the \(2n+1\)-dimensional real Hilbert space of spherical harmonics of degree \(n\) on the \(2\)-dimensional sphere \(S^2.\) Let, for \(f\in\mathcal H_n,\) \(Z(f)=\{x\in S^2:f(x)=0\}\) and \(N(f)\) be the number of connected components of \(Z(f).\)
The authors consider a random gaussian spherical harmonic \(f\) of degree \(n,\) i.e., \(f=\sum_{k=-n}^n\xi_kY_k,\) where \(\xi_k\) are i.i.d. Gaussian random variables with \(E\xi_k^2=\frac{1}{2n+1}\) and \(\{Y_k\}\) is an orthonormal basis of \(\mathcal H_n.\)
The main result of the paper is the following (Theorem 1.1). There exist a constant \(a>0\) such that, for every \(\varepsilon>0,\) we have \[ P\left(\left|\frac{N(f)}{n^2}-a\right|>\varepsilon\right)\leq C(\varepsilon)e^{-c(\varepsilon)n}, \] where \(c(\varepsilon)\) and \(C(\varepsilon)\) are some positive constants depending only on \(\varepsilon.\)