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About large deviations for empirical path measures of cycle counts of integer partitions and their relation to systems of Bosons
Motivated by the Bose gas we introduce certain combinatorial structures. We analyse the asymptotic behaviour of empirical shape measures and of empirical path measures of $ N $ Brownian motions with large deviations techniques. The rate functions are given as variational problems which we analyse. A symmetrised system of Brownian motions, that is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a uniformly distributed random permutation of $1,\dots,N$, is highly correlated and has to be formulated such that standard techniques can be applied. We review a novel spatial and a novel cycle structure approach for the symmetrised distributions of the empirical path measures. The cycle structure leads to a proof of a phase transition in the mean path measure.