Motivated by the Bose gas, this chapter introduces certain combinatorial structures. It analyses 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 that are analysed. A symmetrized system of Brownian motions is highly correlated and has to be formulated such that standard techniques can be applied. The chapter reviews a novel spatial and a novel cycle structure approach for the symmetrized distributions of the empirical path measures. The cycle structure leads to a proof of a phase transition in the mean path measure
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