  Large deviations for fat tailed distributions, i.e. those that decay slower
than exponential, are not only relatively likely, but they also occur in a
rather peculiar way where a finite fraction of the whole sample deviation is
concentrated on a single variable. The regime of large deviations is separated
from the regime of typical fluctuations by a phase transition where the
symmetry between the points in the sample is spontaneously broken. For
stochastic processes with a fat tailed microscopic noise, this implies that
while typical realizations are well described by a diffusion process with
continuous sample paths, large deviation paths are typically discontinuous. For
eigenvalues of random matrices with fat tailed distributed elements, a large
deviation where the trace of the matrix is anomalously large concentrates on
just a single eigenvalue, whereas in the thin tailed world the large deviation
affects the whole distribution. These results find a natural application to
finance. Since the price dynamics of financial stocks is characterized by fat
tailed increments, large fluctuations of stock prices are expected to be
realized by discrete jumps. Interestingly, we find that large excursions of
prices are more likely realized by continuous drifts rather than by
discontinuous jumps. Indeed, auto-correlations suppress the concentration of
large deviations. Financial covariance matrices also exhibit an anomalously
large eigenvalue, the market mode, as compared to the prediction of random
matrix theory. We show that this is explained by a large deviation with excess
covariance rather than by one with excess volatility.
