Gaussian Analytic functions in the polydisk

We study hyperbolic Gaussian analytic functions in the unit polydisk of ${\mathbb{C}}^n$. Following the scheme previously used in the unit ball we first study the asymptotics of fluctuations of linear statistics as the directional intensities $L_j$, $j=1,\dots ,n$ tend to $\mathrm{\infty }$. Then we estimate the probability of large deviations of such linear statistics and use the estimate to prove a hole theorem. Our proofs are inspired by the methods of M. Sodin and B. Tsirelson for the one-dimensional case, and B. Shiffman and S. Zelditch for the study of the analogous problem for compact K"ahler manifolds.