About large deviations for empirical path measures of cycle counts of integer partitions and their relation to systems of Bosons
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Submission date: 27. Sep. 2007
MSC-Numbers: 60F10, 60J65, 82B10, 82B26
Keywords and phrases: large deviations, integer partitions, path measure, Brownian bridges, symmetrised distribution, combinatorial structures
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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 -th motion, where is a uniformly distributed random permutation of , 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.