In this work, starting from 21 months of data from the Fermi Large Area Telescope (LAT), we derive maps of the residual isotropic gamma-ray emission, a relevant fraction of which is expected to be contributed by the extragalactic diffuse gamma-ray background (EGB). We search for the auto-correlation signals in the above gamma-ray maps and for the cross-correlation signal with the angular distribution of different classes of objects that trace the large-scale structure of the Universe. We compute the angular two-point auto-correlation function of the residual Fermi-LAT maps at energies E > 1 GeV, E > 3 GeV and E > 30 GeV well above the Galactic plane and find no significant correlation signal. This is, indeed, what is expected if the EGB were contributed by BL Lacertae (BLLacs), Flat Spectrum Radio Quasars (FSRQs) or star-forming galaxies, since, in this case, the predicted signal is very weak. Then, we search for the Integrated Sachs-Wolfe (ISW) signature by cross-correlating the Fermi-LAT maps with the 7-year Wilkinson Microwave Anisotropy Probe (WMAP7) cosmic microwave background map. We find a cross-correlation consistent with zero, even though the expected signal is larger than that of the EGB auto-correlation. Finally, in an attempt to constrain the nature of the gamma-ray background, we cross-correlate the Fermi-LAT maps with the angular distributions of objects that may contribute to the EGB: quasi-stellar objects (QSOs) in the Sloan Digital Sky Survey Data Release 6 (SDSS-DR6) catalogue, NRAO VLA Sky Survey (NVSS) galaxies, Two Micron All Sky Survey (2MASS) galaxies and Luminous Red Galaxies (LRGs) in the SDSS catalogue. The cross-correlation is always consistent with zero, in agreement with theoretical expectations, but we find (with low statistical significance) some interesting features that may indicate that some specific classes of objects contribute to the EGB. A chi(2) analysis confirms that the correlation properties of the 21-month data do not provide strong constraints of the EGB origin. However, the results suggest that the situation will significantly improve with the 5- and 10-yr Fermi-LAT data. In future, the EGB analysis will then allow placing significant constraints on the nature of the EGB and might provide, in addition, a detection of the ISW signal.
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