  We study large deviations principles for $ N $ random processes on the
lattice $ \Z^d $ with finite time horizon $ [0,\beta] $ under a symmetrised
measure where all initial and terminal points are uniformly given by a random
permutation. That is, given a permutation $ \sigma $ of $ N $ elements and a
vector $ (x_1,...,x_N) $ of $ N $ initial points we let the random processes
terminate in the points $ (x_{\sigma(1)},...,x_{\sigma(N)}) $ and then sum over
all possible permutations and initial points, weighted with an initial
distribution. There is a two-level random mechanism and we prove two-level
large deviations principles for the mean of empirical path measures, for the
mean of paths and for the mean of occupation local times under this symmetrised
measure. The symmetrised measure cannot be written as any product of single
random process distributions. We show a couple of important applications of
these results in quantum statistical mechanics using the Feynman-Kac formulae
representing traces of certain trace class operators. In particular we prove a
non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein
statistics and mean field interactions.
  A special case of our large deviations principle for the mean of occupation
local times of $ N $ simple random walks has the Donsker-Varadhan rate function
as the rate function for the limit $ N\to\infty $ but for finite time $ \beta
$. We give an interpretation in quantum statistical mechanics for this
surprising result.
