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

Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks