  We study the three-dimensional distribution of matter at z~2 using high
resolution spectra of QSO pairs and simulated spectra drawn from cosmological
hydro-dynamical simulations. We present a sample of 15 QSOs, corresponding to
21 baselines of angular separations evenly distributed between ~1 and 14
arcmin, observed with the Ultraviolet and Visual Echelle Spectrograph (UVES) at
the European Southern Observatory-Very Large Telescope (ESO-VLT). The observed
correlation functions of the transmitted flux in the HI Lya forest transverse
to and along the line of sight are in agreement, implying that the distortions
in redshift space due to peculiar velocities are relatively small and - within
the relatively large error bars - not significant. The clustering signal is
significant up to velocity separations of ~300 km/s, corresponding to about 5
h^{-1} comoving Mpc. Compatibility at the 2 sigma level has been found both for
the Auto- and Cross-correlation functions and for the set of the Cross
correlation coefficients. The analysis focuses in particular on two QSO groups
of the sample. Searching for alignments in the redshift space between Lya
absorption lines belonging to different lines of sight, it has been possible to
discover the presence of a wide HI structures extending over about ten Mpc in
comoving space, and give constraints on the sizes of two cosmic under-dense
regions in the intergalactic medium.
