  The Weak Turbulence Theory is a statistical framework to describe a large
ensemble of nonlinearly interacting waves. The archetypal example of such
system is the ocean surface that is made of interacting surface gravity waves.
Here we describe a laboratory experiment dedicated to probe the statistical
properties of turbulent gravity waves. We setup an isotropic state of
interacting gravity waves in the Coriolis facility (13~m diameter circular wave
tank) by exciting waves at 1~Hz by wedge wavemakers. We implement a
stereoscopic technique to obtain a measurement of the surface elevation that is
resolved both in space and time. Fourier analysis shows that the laboratory
spectra are systematically steeper than the theoretical predictions and than
the field observations in the Black Sea by Leckler {\it et al. JPO} 2015. We
identify a strong impact of surface dissipation on the scaling of the Fourier
spectrum at the scales that are accessible in the experiments. We use
bicoherence and tricoherence statistical tools in frequency and/or wavevector
space to identify the active nonlinear coupling. These analyses are also
performed on the field data by Leckler {\it et al.} for comparison with the
laboratory data. 3-wave coupling are characterized and shown to involve mostly
quasi resonances of waves with second or higher order harmonics. 4-wave
coupling are not observed in the laboratory but are evidenced in the field
data. We finally discuss temporal scale separation to explain our observations.
