  When light travels through strongly scattering media with optical gain, the
synergy between diffusive transport and stimulated emission can lead to lasing
action. Below the threshold pump power, the emission spectrum is smooth and
consistent from shot-to-shot. Above the lasing threshold, the spectrum of
emitted light becomes spiky and shows strong fluctuations from shot-to-shot.
Recent experiments have reported that emitted intensity resembles a power-law
distribution (\emph{i.e.} L\'evy statistics). Recent theories have described
the emergence of L\'evy statistics as an intrinsic property of lasing in random
media. To separate intrinsic intensity fluctuations from the motion of
scatterers, we compare the statistics of samples with stationary or
freely-diffusing scatterers. Consistent with previous reports, we observe
L\'evy-like statistics when intensity data are pooled across an ensemble of
scatterer configurations. For fixed scatterers, we find exponential intensity
distributions whose mean intensities vary widely across wavelengths.
L\'evy-like statistics re-emerges when data are combined across many lasing
modes. Additionally, we find strong correlations of lasing peak intensities
across wavelengths. A simple mean-field statistical model captures the observed
one- and two-point statistics, where correlations in emission intensity arise
from competition among all lasing modes for limited gain.
