Asymptotic Feynman-Kac Formulae for Large Symmetrised Systems of Random walks
Stefan Adams and Tony Dorlas
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Submission date: 11. Oct. 2006
published in: Annales de l'Institut Henri Poincaré / B, 44 (2008) 5, p. 837-875
DOI number (of the published article): 10.1214/07-AIHP132
MSC-Numbers: 60F10, 60J65, 82B10, 82B26
Keywords and phrases: large deviations, Bose-Einstein statistics, Feynman-Kac formula, Quantum spin systems
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We study large deviations principles for N random processes on the lattice with finite time horizon under a symmetrised measure where all initial and terminal points are uniformly given by a random permutation. That is, given a permutation of N elements and a vector of N initial points we let the random processes terminate in the points 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 but for finite time . We give an interpretation in quantum statistical mechanics for this surprising result.