  In ultra-fast multi-mode lasers, mode-locking is implemented by means of ad
hoc devices, like saturable absorbers or modulators, allowing for very short
pulses. This comes about because of nonlinear interactions induced among modes
at different, well equispaced, frequencies. Theory predicts that the same
locking of modes would occur in random lasers but, in absence of any device,
its detection is unfeasible so far. Because of the general interest in the
phenomenology and understanding of random lasers and, moreover, because it is a
first example of self-starting mode-locking we devise and test a way to measure
such peculiar non-linear coupling. Through a detailed analysis of multi-mode
correlations we provide clear evidence for the occurrence of nonlinear
mode-coupling in the cavity-less random laser made of a powder of GaAs crystals
and its self-starting mode-locking nature. The behavior of multi-point
correlations among intensity peaks is tested against the nonlinear frequency
matching condition equivalent to the one underlying phase-locking in ordered
ultrafast lasers. Non-trivially large multi-point correlations are clearly
observed for spatially overlapping resonances and turn out to sensitively
depend on the frequency matching being satisfied, eventually demonstrating the
occurrence of non-linear mode-locked mode-coupling.
