  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 $\infty$. 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.
