  Fluctuations play a critical role in cosmology. They are relevant across a
range of phenomena from the dynamics of inflation to the formation of
structure. In many cases, these fluctuations are coarse grained and follow a
Gaussian distribution as a consequence of the Central Limit Theorem. Yet, some
classes of observables are dominated by rare fluctuations and are sensitive to
the details of the underlying microphysics. In this paper, we argue that the
Large Deviation Principle can be used to diagnose when one must to appeal to
the fundamental description. Concretely, we investigate the regime of validity
for the Fokker-Planck equation that governs Stochastic Inflation. For typical
fluctuations, this framework leads to the central limit-type behavior expected
of a random walk. However, fluctuations in the regime of the Large Deviation
Principle are determined by instanton-like saddle points accompanied by a new
energy scale. When this energy scale is above the UV cutoff of the EFT, the
tail is only calculable in the microscopic description. We explicitly
demonstrate this phenomenon in the context of determining the phase transition
to eternal inflation, the distribution of scalar field fluctuations in de
Sitter, and the production of primordial black holes.
