  We have carried out numerical simulations of shot noise in 2D arrays of
single-electron islands with random background charges. The results show that
in contrast with the 1D arrays, at low currents the current noise is strongly
colored, and its spectral density levels off at very low frequencies. The Fano
factor may be much larger than unity, due to the remnants of
single-electron/hole avalanches. However, even very small thermal fluctuations
reduce the Fano factor below 1 for almost any bias.
