  Dense cold atomic clouds have been shown to be similar to plasmas. Previous
studies showed that such clouds exhibit instabilities induced by long-range
interactions. However they did not describe the spatial properties of the
dynamics. In this Letter, we study experimentally the spatial nature of
stochastic instabilities and find out that the dynamics is localized. Data are
analyzed both in the spectral domain and in the spatial domain (principal
component analysis). Both methods fail to describe the dynamics in terms of
eigenmodes, showing that space and time are not separable.
