We investigate the spatial clustering of X-ray selected sources in the two deepest X-ray fields to date, namely the 2Msec Chandra Deep Field North (CDFN) and the 1Msec Chandra Deep Field South (CDFS). The projected correlation function w(r_p), measured on scales ~0.2-10 h^-1 Mpc for a sample of 240 sources with spectroscopic redshift in the CDFN and 124 sources in the CDFS at a median redshift of z~0.8, is used to constrain the amplitude and slope of the real space correlation function xi(r)=(r/r0)^-gamma. The clustering signal is detected at high confidence (>~ 7 sigma) in both fields. The amplitude of the correlation is found to be significantly different in the two fields, the correlation length r0 being 8.6 +- 1.2 h^-1 Mpc in the CDFS and 4.2 +- 0.4 h^-1 Mpc in the CDFN, while the correlation slope gamma is found to be flat in both fields: gamma=1.33 +- 0.11 in the CDFS and gamma=1.42 +- 0.07 in the CDFN (a flat Universe with Omega_m=0.3 and Omega_L=0.7 is assumed; 1 sigma Poisson error estimates are considered). The correlation function has been also measured separately for sources classified as AGN or galaxies. In both fields AGN have a median redshift of z~0.9 and a median 0.5-10 keV luminosity of L_x~10^43 erg s^-1, i.e. they are generally in the Seyfert luminosity regime. As in the case of the total samples, we found a significant difference in the AGN clustering amplitude between the two fields, the best fit correlation parameters being r0=10.3 +- 1.7 h^-1 Mpc, gamma=1.33 +- 0.14 in the CDFS, and r0=5.5 +- 0.6 h^-1 Mpc, gamma=1.50 +- 0.12 in the CDFN. Within each field no statistically significant difference is found between soft and hard X-ray selected sources or between type 1 and type 2 AGN. (abridged)
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