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Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks

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Adams, S. (Stefan) and Dorlas, Tony (2008) Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks. Annales de l'Institut Henri Poincare (B) Probability and Statistics, Vol.44 (No.5). pp. 837-875. doi:10.1214/07-AIHP132 ISSN 0246-0203.

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Official URL: http://dx.doi.org/10.1214/07-AIHP132

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Abstract

We study large deviations principles for N random processes on the lattice ℤd with finite time horizon [0, β] under a symmetrised measure where all initial and terminal points are uniformly averaged over random permutations. That is, given a permutation σ of N elements and a vector (x1, …, xN) of N initial points we let the random processes terminate in the points (xσ(1), …, xσ(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 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 N→∞ but for finite time β. We give an interpretation in quantum statistical mechanics for this surprising result.

Item Type: Journal Article
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title: Annales de l'Institut Henri Poincare (B) Probability and Statistics
Publisher: Institute of Mathematical Statistics
ISSN: 0246-0203
Official Date: October 2008
Dates:
DateEvent
October 2008Published
Volume: Vol.44
Number: No.5
Page Range: pp. 837-875
DOI: 10.1214/07-AIHP132
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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