期刊
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES
卷 44, 期 5, 页码 837-875出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/07-AIHP132
关键词
Large deviations; Large systems of random processes with symmetrised initial-terminal conditions; Feynman-Kac formula; Bose-Einstein statistics; Non-commutative Varadhan lemma; Quantum spin systems; Donsker-Varadhan function
资金
- DFG [AD 194/1-3]
We study large deviations principles for N random processes on the lattice Z(d) with finite time horizon [0, beta] under a symmetrised measure where all initial and terminal points are uniformly averaged over random permutations. That is, given a permutation a of N elements and a vector (x(1), . . . , x(N)) of N initial points we let the random processes terminate in the points (x sigma(1), . . . , x(sigma(N))) and then sum over all possible permutations and initial points, weighted with an initial distribution. We prove level-two 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 a product of single random process distributions. We show I 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 N -> infinity but for finite time. We give an interpretation in quantum statistical mechanics for this surprising result.
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